cons_cumulative.c

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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
/*                                                                           */
/*                  This file is part of the program and library             */
/*         SCIP --- Solving Constraint Integer Programs                      */
/*                                                                           */
/*    Copyright (C) 2002-2015 Konrad-Zuse-Zentrum                            */
/*                            fuer Informationstechnik Berlin                */
/*                                                                           */
/*  SCIP is distributed under the terms of the ZIB Academic License.         */
/*                                                                           */
/*  You should have received a copy of the ZIB Academic License              */
/*  along with SCIP; see the file COPYING. If not email to scip@zib.de.      */
/*                                                                           */
/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */

/**@file   cons_cumulative.c
 * @brief  constraint handler for cumulative constraints
 * @author Timo Berthold
 * @author Stefan Heinz
 * @author Jens Schulz
 *
 * Given:
 * - a set of jobs, represented by their integer start time variables \f$S_j\f$, their array of processing times \f$p_j\f$ and of
 *   their demands \f$d_j\f$.
 * - an integer resource capacity \f$C\f$
 *
 * The cumulative constraint ensures that for each point in time \f$t\f$ \f$\sum_{j: S_j \leq t < S_j + p_j} d_j \leq C\f$ holds.
 *
 * Separation:
 * - can be done using binary start time model, see Pritskers, Watters and Wolfe
 * - or by just separating relatively weak cuts on the integer start time variables
 *
 * Propagation:
 * - time tabling, Klein & Scholl (1999)
 * - Edge-finding from Petr Vilim, adjusted and simplified for dynamic repropagation
 *   (2009)
 * - energetic reasoning, see Baptiste, Le Pape, Nuijten (2001)
 *
 */

/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/

#include <assert.h>
#include <string.h>

#include "tclique/tclique.h"
#include "scip/cons_cumulative.h"
#include "scip/cons_linking.h"
#include "scip/cons_knapsack.h"
#include "scip/scipdefplugins.h"

/**@name Constraint handler properties
 *
 * @{
 */

/* constraint handler properties */
#define CONSHDLR_NAME          "cumulative"
#define CONSHDLR_DESC          "cumulative constraint handler"
#define CONSHDLR_SEPAPRIORITY   2100000 /**< priority of the constraint handler for separation */
#define CONSHDLR_ENFOPRIORITY  -2040000 /**< priority of the constraint handler for constraint enforcing */
#define CONSHDLR_CHECKPRIORITY -3030000 /**< priority of the constraint handler for checking feasibility */
#define CONSHDLR_SEPAFREQ             1 /**< frequency for separating cuts; zero means to separate only in the root node */
#define CONSHDLR_PROPFREQ             1 /**< frequency for propagating domains; zero means only preprocessing propagation */
#define CONSHDLR_EAGERFREQ          100 /**< frequency for using all instead of only the useful constraints in separation,
                                         *   propagation and enforcement, -1 for no eager evaluations, 0 for first only */
#define CONSHDLR_MAXPREROUNDS        -1 /**< maximal number of presolving rounds the constraint handler participates in (-1: no limit) */
#define CONSHDLR_DELAYSEPA        FALSE /**< should separation method be delayed, if other separators found cuts? */
#define CONSHDLR_DELAYPROP        FALSE /**< should propagation method be delayed, if other propagators found reductions? */
#define CONSHDLR_NEEDSCONS         TRUE /**< should the constraint handler be skipped, if no constraints are available? */

#define CONSHDLR_PRESOLTIMING            SCIP_PRESOLTIMING_ALWAYS
#define CONSHDLR_PROP_TIMING             SCIP_PROPTIMING_BEFORELP

/**@} */

/**@name Default parameter values
 *
 * @{
 */

/* default parameter values */

/* separation */
#define DEFAULT_USEBINVARS             FALSE /**< should the binary representation be used? */
#define DEFAULT_LOCALCUTS              FALSE /**< should cuts be added only locally? */
#define DEFAULT_USECOVERCUTS            TRUE /**< should covering cuts be added? */
#define DEFAULT_CUTSASCONSS             TRUE /**< should the cuts be created as knapsack constraints? */
#define DEFAULT_SEPAOLD                 TRUE /**< shall old sepa algo be applied? */

/* propagation */
#define DEFAULT_TTINFER                 TRUE /**< should time-table (core-times) propagator be used to infer bounds? */
#define DEFAULT_EFCHECK                FALSE /**< should edge-finding be used to detect an overload? */
#define DEFAULT_EFINFER                FALSE /**< should edge-finding be used to infer bounds? */
#define DEFAULT_USEADJUSTEDJOBS        FALSE /**< should during edge-finding jobs be adusted which run on the border of the effective time horizon? */
#define DEFAULT_TTEFCHECK               TRUE /**< should time-table edge-finding be used to detect an overload? */
#define DEFAULT_TTEFINFER               TRUE /**< should time-table edge-finding be used to infer bounds? */

/* presolving */
#define DEFAULT_DUALPRESOLVE            TRUE /**< should dual presolving be applied? */
#define DEFAULT_COEFTIGHTENING         FALSE /**< should coeffisient tightening be applied? */
#define DEFAULT_NORMALIZE               TRUE /**< should demands and capacity be normalized? */
#define DEFAULT_PRESOLPAIRWISE          TRUE /**< should pairwise constraint comparison be performed in presolving? */
#define DEFAULT_DISJUNCTIVE             TRUE /**< extract disjunctive constraints? */
#define DEFAULT_DETECTDISJUNCTIVE       TRUE /**< search for conflict set via maximal cliques to detect disjunctive constraints */
#define DEFAULT_DETECTVARBOUNDS         TRUE /**< search for conflict set via maximal cliques to detect variable bound constraints */
#define DEFAULT_MAXNODES             10000LL /**< number of branch-and-bound nodes to solve an independent cumulative constraint  (-1: no limit) */

/* enforcement */
#define DEFAULT_FILLBRANCHCANDS        FALSE /**< should branching candidates be added to storage? */

/* conflict analysis */
#define DEFAULT_USEBDWIDENING           TRUE /**< should bound widening be used during conflict analysis? */

/**@} */

/**@name Event handler properties
 *
 * @{
 */

#define EVENTHDLR_NAME         "cumulative"
#define EVENTHDLR_DESC         "bound change event handler for cumulative constraints"

/**@} */

/*
 * Data structures
 */

/** constraint data for cumulative constraints */
struct SCIP_ConsData
{
   SCIP_VAR**            vars;               /**< array of variable representing the start time of each job */
   SCIP_Bool*            downlocks;          /**< array to store if the variable has a down lock */
   SCIP_Bool*            uplocks;            /**< array to store if the variable has an uplock */
   SCIP_CONS**           linkingconss;       /**< array of linking constraints for the integer variables */
   SCIP_ROW**            demandrows;         /**< array of rows of linear relaxation of this problem */
   SCIP_ROW**            scoverrows;         /**< array of rows of small cover cuts of this problem */
   SCIP_ROW**            bcoverrows;         /**< array of rows of big cover cuts of this problem */
   int*                  demands;            /**< array containing corresponding demands */
   int*                  durations;          /**< array containing corresponding durations */
   SCIP_Real             resstrength1;       /**< stores the resource strength 1*/
   SCIP_Real             resstrength2;       /**< stores the resource strength 2 */
   SCIP_Real             cumfactor1;         /**< stroes the cumulativeness of the constraint */
   SCIP_Real             disjfactor1;        /**< stores the disjunctiveness of the constraint */
   SCIP_Real             disjfactor2;        /**< stores the disjunctiveness of the constraint */
   SCIP_Real             estimatedstrength;
   int                   nvars;              /**< number of variables */
   int                   varssize;           /**< size of the arrays */
   int                   ndemandrows;        /**< number of rows of cumulative constrint for linear relaxation */
   int                   demandrowssize;     /**< size of array rows of demand rows */
   int                   nscoverrows;        /**< number of rows of small cover cuts */
   int                   scoverrowssize;     /**< size of array of small cover cuts */
   int                   nbcoverrows;        /**< number of rows of big cover cuts */
   int                   bcoverrowssize;     /**< size of array of big cover cuts */
   int                   capacity;           /**< available cumulative capacity */

   int                   hmin;               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax;               /**< right bound of time axis to be considered  (not including hmax) */

   unsigned int          signature;          /**< constraint signature which is need for pairwise comparison */

   unsigned int          validsignature:1;   /**< is the signature valid */
   unsigned int          normalized:1;       /**< is the constraint normalized */
   unsigned int          covercuts:1;        /**< cover cuts are created? */
   unsigned int          propagated:1;       /**< is constraint propagted */
   unsigned int          varbounds:1;        /**< bool to store if variable bound strengthening was already preformed */
   unsigned int          triedsolving:1;     /**< bool to store if we tried already to solve that constraint as independent subproblem */

#ifdef SCIP_STATISTIC
   int                   maxpeak;
#endif
};

/** constraint handler data */
struct SCIP_ConshdlrData
{
   SCIP_EVENTHDLR*       eventhdlr;          /**< event handler for bound change events */

   SCIP_Bool             usebinvars;         /**< should the binary variables be used? */
   SCIP_Bool             cutsasconss;        /**< should the cumulative constraint create cuts as knapsack constraints? */
   SCIP_Bool             ttinfer;            /**< should time-table (core-times) propagator be used to infer bounds? */
   SCIP_Bool             efcheck;            /**< should edge-finding be used to detect an overload? */
   SCIP_Bool             efinfer;            /**< should edge-finding be used to infer bounds? */
   SCIP_Bool             useadjustedjobs;    /**< should during edge-finding jobs be adusted which run on the border of the effective time horizon? */
   SCIP_Bool             ttefcheck;          /**< should time-table edge-finding be used to detect an overload? */
   SCIP_Bool             ttefinfer;          /**< should time-table edge-finding be used to infer bounds? */
   SCIP_Bool             localcuts;          /**< should cuts be added only locally? */
   SCIP_Bool             usecovercuts;       /**< should covering cuts be added? */
   SCIP_Bool             sepaold;            /**< shall old sepa algo be applied? */


   SCIP_Bool             fillbranchcands;    /**< should branching candidates be added to storage? */

   SCIP_Bool             dualpresolve;       /**< should dual presolving be applied? */
   SCIP_Bool             coeftightening;     /**< should coeffisient tightening be applied? */
   SCIP_Bool             normalize;          /**< should demands and capacity be normalized? */
   SCIP_Bool             disjunctive;        /**< extract disjunctive constraints? */
   SCIP_Bool             detectdisjunctive;  /**< search for conflict set via maximal cliques to detect disjunctive constraints */
   SCIP_Bool             detectvarbounds;    /**< search for conflict set via maximal cliques to detect variable bound constraints */
   SCIP_Bool             usebdwidening;      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool             presolpairwise;     /**< should pairwise constraint comparison be performed in presolving? */
   SCIP_Bool             detectedredundant;  /**< was detection of redundant constraints already performed? */

   SCIP_Longint          maxnodes;           /**< number of branch-and-bound nodes to solve an independent cumulative constraint  (-1: no limit) */

   SCIP_DECL_SOLVECUMULATIVE((*solveCumulative)); /**< method to use a single cumulative condition */

   /* statistic values which are collected if SCIP_STATISTIC is defined */
#ifdef SCIP_STATISTIC
   SCIP_Longint          nlbtimetable;       /**< number of times the lower bound was tightened by the time-table propagator */
   SCIP_Longint          nubtimetable;       /**< number of times the upper bound was tightened by the time-table propagator */
   SCIP_Longint          ncutofftimetable;   /**< number of times the a cutoff was detected due to time-table propagator */
   SCIP_Longint          nlbedgefinder;      /**< number of times the lower bound was tightened by the edge-finder propagator */
   SCIP_Longint          nubedgefinder;      /**< number of times the upper bound was tightened by the edge-finder propagator */
   SCIP_Longint          ncutoffedgefinder;  /**< number of times the a cutoff was detected due to edge-finder propagator */
   SCIP_Longint          ncutoffoverload;    /**< number of times the a cutoff was detected due to overload checking via edge-finding */
   SCIP_Longint          nlbTTEF;            /**< number of times the lower bound was tightened by time-table edge-finding */
   SCIP_Longint          nubTTEF;            /**< number of times the upper bound was tightened by time-table edge-finding */
   SCIP_Longint          ncutoffoverloadTTEF;/**< number of times the a cutoff was detected due to overload checking via time-table edge-finding */

   int                   nirrelevantjobs;    /**< number of time a irrelevant/redundant jobs was removed form a constraint */
   int                   nalwaysruns;        /**< number of time a job removed form a constraint which run completely during the effective horizon */
   int                   nremovedlocks;      /**< number of times a up or down lock was removed */
   int                   ndualfixs;          /**< number of times a dual fix was performed by a single constraint */
   int                   ndecomps;           /**< number of times a constraint was decomposed */
   int                   ndualbranchs;       /**< number of times a dual branch was discoverd and applicable via probing */
   int                   nallconsdualfixs;   /**< number of times a dual fix was performed due to knowledge of all cumulative constraints */
   int                   naddedvarbounds;    /**< number of added variable bounds constraints */
   int                   naddeddisjunctives; /**< number of added disjunctive constraints */

   SCIP_Bool             iscopy;             /**< Boolean to store if constraint handler is part of a copy */
#endif
};

/**@name Inference Information Methods
 *
 *  An inference information can be passed with each domain reduction to SCIP. This information is passed back to the
 *  constraint handler if the corresponding bound change has to be explained. It can be used to store information which
 *  help to construct a reason/explanation for a bound change. The inference information is limited to size of integer.
 *
 *  In case of the cumulative constraint handler we store the used propagation algorithms for that particular bound
 *  change and the earliest start and latest completion time of all jobs in the conflict set.
 *
 * @{
 */

/** Propagation rules */
enum Proprule
{
   PROPRULE_1_CORETIMES          = 1,        /**< core-time propagator */
   PROPRULE_2_EDGEFINDING        = 2,        /**< edge-finder */
   PROPRULE_3_TTEF               = 3         /**< time-table edeg-finding */
};
typedef enum Proprule PROPRULE;

/** inference information */
struct InferInfo
{
   union
   {
      /** struct to use the inference information */
      struct
      {
         unsigned int    proprule:2;         /**< propagation rule that was applied */
         unsigned int    data1:15;           /**< data field one */
         unsigned int    data2:15;           /**< data field two */
      } asbits;
      int                asint;              /**< inference information as a single int value */
   } val;
};
typedef struct InferInfo INFERINFO;

/** converts an integer into an inference information */
static
INFERINFO intToInferInfo(
   int                   i                   /**< integer to convert */
   )
{
   INFERINFO inferinfo;

   inferinfo.val.asint = i;

   return inferinfo;
}

/** converts an inference information into an int */
static
int inferInfoToInt(
   INFERINFO             inferinfo           /**< inference information to convert */
   )
{
   return inferinfo.val.asint;
}

/** returns the propagation rule stored in the inference information */
static
PROPRULE inferInfoGetProprule(
   INFERINFO             inferinfo           /**< inference information to convert */
   )
{
   return (PROPRULE) inferinfo.val.asbits.proprule;
}

/** returns data field one of the inference information */
static
int inferInfoGetData1(
   INFERINFO             inferinfo           /**< inference information to convert */
   )
{
   return (int) inferinfo.val.asbits.data1;
}

/** returns data field two of the inference information */
static
int inferInfoGetData2(
   INFERINFO             inferinfo           /**< inference information to convert */
   )
{
   return (int) inferinfo.val.asbits.data2;
}


/** constructs an inference information out of a propagation rule, an earliest start and a latest completion time */
static
INFERINFO getInferInfo(
   PROPRULE              proprule,           /**< propagation rule that deduced the value */
   int                   data1,              /**< data field one */
   int                   data2               /**< data field two */
   )
{
   INFERINFO inferinfo;

   /* check that the data menber are in the range of the available bits */
   assert((int)proprule <= (1<<2)-1); /*lint !e685*/
   assert(data1 >= 0 && data1 < (1<<15));
   assert(data2 >= 0 && data2 < (1<<15));

   inferinfo.val.asbits.proprule = proprule; /*lint !e641*/
   inferinfo.val.asbits.data1 = (unsigned int) data1; /*lint !e732*/
   inferinfo.val.asbits.data2 = (unsigned int) data2; /*lint !e732*/

   return inferinfo;
}

/**@} */

/*
 * Local methods
 */

/**@name Miscellaneous Methods
 *
 * @{
 */

#ifndef NDEBUG

/** compute the core of a job which lies in certain interval [begin, end) */
static
int computeCoreWithInterval(
   int                   begin,              /**< begin of the interval */
   int                   end,                /**< end of the interval */
   int                   ect,                /**< earliest completion time */
   int                   lst                 /**< latest start time */
   )
{
   int core;

   core = MAX(0, MIN(end, ect) - MAX(lst, begin));

   return core;
}
#else
#define computeCoreWithInterval(begin, end, ect, lst) (MAX(0, MIN((end), (ect)) - MAX((lst), (begin))))
#endif

/** returns the implied earliest start time */
static
SCIP_RETCODE computeImpliedEst(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< variable for which the implied est should be returned */
   SCIP_HASHMAP*         addedvars,          /**< hash map containig the variable which are already added */
   int*                  est                 /**< pointer to store the implied earliest start time */
   )
{  /*lint --e{715}*/
#if 0
   SCIP_VAR** vbdvars;
   SCIP_VAR* vbdvar;
   SCIP_Real* vbdcoefs;
   SCIP_Real* vbdconsts;
   void* image;
   int nvbdvars;
   int v;
#endif

   (*est) = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));

#if 0
   /* the code contains a bug; we need to check if an implication forces that the jobs do not run in parallel */

   nvbdvars = SCIPvarGetNVlbs(var);
   vbdvars = SCIPvarGetVlbVars(var);
   vbdcoefs = SCIPvarGetVlbCoefs(var);
   vbdconsts = SCIPvarGetVlbConstants(var);

   for( v = 0; v < nvbdvars; ++v )
   {
      vbdvar = vbdvars[v];
      assert(vbdvar != NULL);

      image = SCIPhashmapGetImage(addedvars, (void*)vbdvar);

      if( image != NULL && SCIPisEQ(scip, vbdcoefs[v], 1.0 ) )
      {
         int duration;
         int vbdconst;

         duration = (int)(size_t)image;
         vbdconst = SCIPconvertRealToInt(scip, vbdconsts[v]);

         SCIPdebugMessage("check implication <%s>[%g,%g] >= <%s>[%g,%g] + <%g>\n",
            SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var),
            SCIPvarGetName(vbdvar), SCIPvarGetLbLocal(vbdvar), SCIPvarGetUbLocal(vbdvar), vbdconsts[v]);

         if( duration >= vbdconst )
         {
            int impliedest;

            impliedest =  SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vbdvar)) + duration;

            if( (*est) < impliedest )
            {
               (*est) = impliedest;

               SCIP_CALL( SCIPhashmapRemove(addedvars, (void*)vbdvar) );
            }
         }
      }
   }
#endif

   return SCIP_OKAY;
}

/** returns the implied latest completion time */
static
SCIP_RETCODE computeImpliedLct(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< variable for which the implied est should be returned */
   int                   duration,           /**< duration of the given job */
   SCIP_HASHMAP*         addedvars,          /**< hash map containig the variable which are already added */
   int*                  lct                 /**< pointer to store the implied latest completion time */
   )
{  /*lint --e{715}*/
#if 0
   SCIP_VAR** vbdvars;
   SCIP_VAR* vbdvar;
   SCIP_Real* vbdcoefs;
   SCIP_Real* vbdconsts;
   int nvbdvars;
   int v;
#endif

   (*lct) = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration;

#if 0
   /* the code contains a bug; we need to check if an implication forces that the jobs do not run in parallel */

   nvbdvars = SCIPvarGetNVubs(var);
   vbdvars = SCIPvarGetVubVars(var);
   vbdcoefs = SCIPvarGetVubCoefs(var);
   vbdconsts = SCIPvarGetVubConstants(var);

   for( v = 0; v < nvbdvars; ++v )
   {
      vbdvar = vbdvars[v];
      assert(vbdvar != NULL);

      if( SCIPhashmapExists(addedvars, (void*)vbdvar) &&  SCIPisEQ(scip, vbdcoefs[v], 1.0 ) )
      {
         int vbdconst;

         vbdconst = SCIPconvertRealToInt(scip, -vbdconsts[v]);

         SCIPdebugMessage("check implication <%s>[%g,%g] <= <%s>[%g,%g] + <%g>\n",
            SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var),
            SCIPvarGetName(vbdvar), SCIPvarGetLbLocal(vbdvar), SCIPvarGetUbLocal(vbdvar), vbdconsts[v]);

         if( duration >= -vbdconst )
         {
            int impliedlct;

            impliedlct = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vbdvar));

            if( (*lct) > impliedlct )
            {
               (*lct) = impliedlct;

               SCIP_CALL( SCIPhashmapRemove(addedvars, (void*)vbdvar) );
            }
         }
      }
   }
#endif

   return SCIP_OKAY;
}

/** collects all necessary binary variables to represent the jobs which can be active at time point of interest */
static
SCIP_RETCODE collectBinaryVars(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   SCIP_VAR***           vars,               /**< pointer to the array to store the binary variables */
   int**                 coefs,              /**< pointer to store the coefficients */
   int*                  nvars,              /**< number if collect binary variables */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished           /**< number of jobs that finished before curtime or at curtime */
   )
{
   int nrowvars;
   int startindex;
   int size;

   size = 10;
   nrowvars = 0;
   startindex = nstarted - 1;

   SCIP_CALL( SCIPallocBufferArray(scip, vars, size) );
   SCIP_CALL( SCIPallocBufferArray(scip, coefs, size) );

   /* search for the (nstarted - nfinished) jobs which are active at curtime */
   while( nstarted - nfinished > nrowvars )
   {
      SCIP_VAR* var;
      int endtime;
      int duration;
      int demand;
      int varidx;

      /* collect job information */
      varidx = startindices[startindex];
      assert(varidx >= 0 && varidx < consdata->nvars);

      var = consdata->vars[varidx];
      duration = consdata->durations[varidx];
      demand = consdata->demands[varidx];
      assert(var != NULL);

      endtime = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var)) + duration;

      /* check the end time of this job is larger than the curtime; in this case the job is still running */
      if( endtime > curtime )
      {
         SCIP_VAR** binvars;
         int* vals;
         int nbinvars;
         int start;
         int end;
         int b;

         /* check if the linking constraints exists */
         assert(SCIPexistsConsLinking(scip, var));
         assert(SCIPgetConsLinking(scip, var) != NULL);
         assert(SCIPgetConsLinking(scip, var) == consdata->linkingconss[varidx]);

         /* collect linking constraint information */
         SCIP_CALL( SCIPgetBinvarsLinking(scip, consdata->linkingconss[varidx], &binvars, &nbinvars) );
         vals = SCIPgetValsLinking(scip, consdata->linkingconss[varidx]);

         start = curtime - duration + 1;
         end = MIN(curtime, endtime - duration);

         for( b = 0; b < nbinvars; ++b )
         {
            if( vals[b] < start )
               continue;

            if( vals[b] > end )
               break;

            assert(binvars[b] != NULL);

            /* ensure array proper array size */
            if( size == *nvars )
            {
               size *= 2;
               SCIP_CALL( SCIPreallocBufferArray(scip, vars, size) );
               SCIP_CALL( SCIPreallocBufferArray(scip, coefs, size) );
            }

            (*vars)[*nvars] = binvars[b];
            (*coefs)[*nvars] = demand;
            (*nvars)++;
         }
         nrowvars++;
      }

      startindex--;
   }

   return SCIP_OKAY;
}

/** collect all integer variable which belong to jobs which can run at the point of interest */
static
SCIP_RETCODE collectIntVars(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   SCIP_VAR***           activevars,         /**< jobs that are currently running */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished,          /**< number of jobs that finished before curtime or at curtime */
   SCIP_Bool             lower,              /**< shall cuts be created due to lower or upper bounds? */
   int*                  lhs                 /**< lhs for the new row sum of lbs + minoffset */
   )
{
   SCIP_VAR* var;
   int startindex;
   int endtime;
   int duration;
   int starttime;

   int varidx;
   int sumofstarts;
   int mindelta;
   int counter;

   assert(curtime >= consdata->hmin);
   assert(curtime < consdata->hmax);

   counter = 0;
   sumofstarts = 0;

   mindelta = INT_MAX;

   startindex = nstarted - 1;

   /* search for the (nstarted - nfinished) jobs which are active at curtime */
   while( nstarted - nfinished > counter )
   {
      assert(startindex >= 0);

      /* collect job information */
      varidx = startindices[startindex];
      assert(varidx >= 0 && varidx < consdata->nvars);

      var = consdata->vars[varidx];
      duration = consdata->durations[varidx];
      assert(duration > 0);
      assert(var != NULL);

      if( lower )
         starttime = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      else
         starttime = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      endtime = MIN(starttime + duration, consdata->hmax);

      /* check the end time of this job is larger than the curtime; in this case the job is still running */
      if( endtime > curtime )
      {
         (*activevars)[counter] = var;
         sumofstarts += starttime;
         mindelta = MIN(mindelta, endtime - curtime); /* this amount of schifting holds for lb and ub */
         counter++;
      }

      startindex--;
   }

   assert(mindelta > 0);
   *lhs = lower ? sumofstarts + mindelta : sumofstarts - mindelta;

   return SCIP_OKAY;
}

/** initialize the sorted event point arrays */
static
void createSortedEventpoints(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations per start time variable */
   int*                  starttimes,         /**< array to store sorted start events */
   int*                  endtimes,           /**< array to store sorted end events */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int*                  endindices,         /**< permutation with rspect to the end times */
   SCIP_Bool             local               /**< shall local bounds be used */
   )
{
   SCIP_VAR* var;
   int j;

   assert(vars != NULL || nvars == 0);

   /* assign variables, start and endpoints to arrays */
   for ( j = 0; j < nvars; ++j )
   {
      assert(vars != NULL);

      var = vars[j];
      assert(var != NULL);

      if( local )
         starttimes[j] = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      else
         starttimes[j] = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));

      startindices[j] = j;

      if( local )
         endtimes[j] = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + durations[j];
      else
         endtimes[j] = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var)) + durations[j];

      endindices[j] = j;

   }

   /* sort the arrays not-decreasing according to startsolvalues and endsolvalues (and sort the indices in the same way) */
   SCIPsortIntInt(starttimes, startindices, j);
   SCIPsortIntInt(endtimes, endindices, j);
}

/** initialize the sorted event point arrays w.r.t. the given primal solutions */
static
void createSortedEventpointsSol(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_SOL*             sol,                /**< solution */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations per start time variable */
   int*                  starttimes,         /**< array to store sorted start events */
   int*                  endtimes,           /**< array to store sorted end events */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int*                  endindices          /**< permutation with rspect to the end times */
   )
{
   SCIP_VAR* var;
   int j;

   assert(vars != NULL || nvars == 0);

   /* assign variables, start and endpoints to arrays */
   for ( j = 0; j < nvars; ++j )
   {
      assert(vars != NULL);

      var = vars[j];
      assert(var != NULL);

      starttimes[j] = SCIPconvertRealToInt(scip, SCIPgetSolVal(scip, sol, var));
      startindices[j] = j;

      endtimes[j] = SCIPconvertRealToInt(scip, SCIPgetSolVal(scip, sol, var)) + durations[j];
      endindices[j] = j;

   }

   /* sort the arrays not-decreasing according to startsolvalues and endsolvalues (and sort the indices in the same way) */
   SCIPsortIntInt(starttimes, startindices, j);
   SCIPsortIntInt(endtimes, endindices, j);
}

/** initialize the sorted event point arrays
 *
 * @todo Check the separation process!
 */
static
void createSelectedSortedEventpointsSol(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   SCIP_SOL*             sol,                /**< primal CIP solution, NULL for current LP solution */
   int*                  starttimes,         /**< array to store sorted start events */
   int*                  endtimes,           /**< array to store sorted end events */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int*                  endindices,         /**< permutation with rspect to the end times */
   int*                  nvars,              /**< number of variables that are integral */
   SCIP_Bool             lower               /**< shall the constraints be derived for lower or upper bounds? */
   )
{
   SCIP_VAR* var;
   int tmpnvars;
   int j;

   tmpnvars = consdata->nvars;
   *nvars = 0;

   /* assign variables, start and endpoints to arrays */
   for ( j = 0; j < tmpnvars; ++j )
   {
      var = consdata->vars[j];
      assert(var != NULL);
      assert(consdata->durations[j] > 0);
      assert(consdata->demands[j] > 0);

      if( lower )
      {
         /* only consider jobs that are at their lower or upper bound */
         if( !SCIPisFeasIntegral(scip, SCIPgetSolVal(scip, sol, var))
            || !SCIPisFeasEQ(scip, SCIPgetSolVal(scip, sol, var), SCIPvarGetLbLocal(var)) )
            continue;


         starttimes[*nvars] = SCIPconvertRealToInt(scip, SCIPgetSolVal(scip, sol, var));
         startindices[*nvars] = j;

         endtimes[*nvars] =  starttimes[*nvars] + consdata->durations[j];
         endindices[*nvars] = j;

         SCIPdebugMessage("%d: variable <%s>[%g,%g] (sol %g, duration %d) starttime %d, endtime = %d, demand = %d\n",
            *nvars, SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var), SCIPgetSolVal(scip, sol, var),
            consdata->durations[j],
            starttimes[*nvars], starttimes[*nvars] + consdata->durations[startindices[*nvars]],
            consdata->demands[startindices[*nvars]]);

         (*nvars)++;
      }
      else
      {
         if( !SCIPisFeasIntegral(scip, SCIPgetSolVal(scip, sol, var))
            || !SCIPisFeasEQ(scip, SCIPgetSolVal(scip, sol, var), SCIPvarGetUbLocal(var)) )
            continue;

         starttimes[*nvars] = SCIPconvertRealToInt(scip, SCIPgetSolVal(scip, sol, var));
         startindices[*nvars] = j;

         endtimes[*nvars] =  starttimes[*nvars] + consdata->durations[j];
         endindices[*nvars] = j;

         SCIPdebugMessage("%d: variable <%s>[%g,%g] (sol %g, duration %d) starttime %d, endtime = %d, demand = %d\n",
            *nvars, SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var), SCIPgetSolVal(scip, sol, var),
            consdata->durations[j],
            starttimes[*nvars], starttimes[*nvars] + consdata->durations[startindices[*nvars]],
            consdata->demands[startindices[*nvars]]);

         (*nvars)++;
      }
   }

   /* sort the arrays not-decreasing according to startsolvalues and endsolvalues (and sort the indices in the same way) */
   SCIPsortIntInt(starttimes, startindices, *nvars);
   SCIPsortIntInt(endtimes, endindices, *nvars);

#ifdef SCIP_DEBUG
   SCIPdebugMessage("sorted output %d\n", *nvars);

   for ( j = 0; j < *nvars; ++j )
   {
      SCIPdebugMessage("%d: job[%d] starttime %d, endtime = %d, demand = %d\n", j,
         startindices[j], starttimes[j], starttimes[j] + consdata->durations[startindices[j]],
         consdata->demands[startindices[j]]);
   }

   for ( j = 0; j < *nvars; ++j )
   {
      SCIPdebugMessage("%d: job[%d] endtime %d,  demand = %d\n", j, endindices[j], endtimes[j],
         consdata->demands[endindices[j]]);
   }
#endif
}

#ifdef SCIP_STATISTIC
/** this method checks for relevant intervals for energetic reasoning */
static
SCIP_RETCODE computeRelevantEnergyIntervals(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   int**                 timepoints,         /**< array to store relevant points in time */
   SCIP_Real**           cumulativedemands,  /**< array to store the estimated cumulative demand for each point in time */
   int*                  ntimepoints,        /**< pointer to store the number of timepoints */
   int*                  maxdemand,          /**< pointer to store maximum over all demands */
   SCIP_Real*            minfreecapacity     /**< pointer to store the minimum free capacity */
   )
{
   int* starttimes;         /* stores when each job is starting */
   int* endtimes;           /* stores when each job ends */
   int* startindices;       /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;         /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   SCIP_Real totaldemand;
   int curtime;             /* point in time which we are just checking */
   int endindex;            /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int j;

   assert( scip != NULL );
   assert(durations != NULL);
   assert(demands != NULL);
   assert(capacity >= 0);

   /* if no activities are associated with this cumulative then this constraint is redundant */
   if( nvars == 0 )
      return SCIP_OKAY;

   assert(vars != NULL);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   /* create event point arrays */
   createSortedEventpoints(scip, nvars, vars, durations, starttimes, endtimes, startindices, endindices, TRUE);

   endindex = 0;
   totaldemand = 0.0;

   *ntimepoints = 0;
   (*timepoints)[0] = starttimes[0];
   (*cumulativedemands)[0] = 0;
   *maxdemand = 0;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      int lct;
      int idx;

      curtime = starttimes[j];

      if( curtime >= hmax )
         break;

      /* free all capacity usages of jobs the are no longer running */
      while( endindex < nvars && endtimes[endindex] <= curtime )
      {
         int est;

         if( (*timepoints)[*ntimepoints] < endtimes[endindex] )
         {
            (*ntimepoints)++;
            (*timepoints)[*ntimepoints] = endtimes[endindex];
            (*cumulativedemands)[*ntimepoints] = 0;
         }

         idx = endindices[endindex];
         est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[idx]));
         totaldemand -= (SCIP_Real) demands[idx] * durations[idx] / (endtimes[endindex] - est);
         endindex++;

         (*cumulativedemands)[*ntimepoints] = totaldemand;
      }

      idx = startindices[j];
      lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[idx]) + durations[idx]);
      totaldemand += (SCIP_Real) demands[idx] * durations[idx] / (lct - starttimes[j]);

      if( (*timepoints)[*ntimepoints] < curtime )
      {
         (*ntimepoints)++;
         (*timepoints)[*ntimepoints] = curtime;
         (*cumulativedemands)[*ntimepoints] = 0;
      }

      (*cumulativedemands)[*ntimepoints] = totaldemand;

      /* add the relative capacity requirements for all job which start at the curtime */
      while( j+1 < nvars && starttimes[j+1] == curtime )
      {
         ++j;
         idx = startindices[j];
         lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[idx]) + durations[idx]);
         totaldemand += (SCIP_Real) demands[idx] * durations[idx] / (lct - starttimes[j]);

         (*cumulativedemands)[*ntimepoints] = totaldemand;
      }
   } /*lint --e{850}*/

   /* free all capacity usages of jobs that are no longer running */
   while( endindex < nvars/* && endtimes[endindex] < hmax*/)
   {
      int est;
      int idx;

      if( (*timepoints)[*ntimepoints] < endtimes[endindex] )
      {
         (*ntimepoints)++;
         (*timepoints)[*ntimepoints] = endtimes[endindex];
         (*cumulativedemands)[*ntimepoints] = 0;
      }

      idx = endindices[endindex];
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[idx]));
      totaldemand -= (SCIP_Real) demands[idx] * durations[idx] / (endtimes[endindex] - est);
      (*cumulativedemands)[*ntimepoints] = totaldemand;

      ++endindex;
   }

   (*ntimepoints)++;
   /* compute minimum free capacity */
   (*minfreecapacity) = INT_MAX;
   for( j = 0; j < *ntimepoints; ++j )
   {
      if( (*timepoints)[j] >= hmin && (*timepoints)[j] < hmax )
         *minfreecapacity = MIN( *minfreecapacity, (SCIP_Real)capacity - (*cumulativedemands)[j] );
   }

   /* free buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   return SCIP_OKAY;
}

/** evaluates the cumulativeness and disjointness factor of a cumulative constraint */
static
SCIP_RETCODE evaluateCumulativeness(
   SCIP*                 scip,               /**< pointer to scip */
   SCIP_CONS*            cons                /**< cumulative constraint */
   )
{
   SCIP_CONSDATA* consdata;
   int nvars;
   int v;
   int capacity;

   /* output values: */
   SCIP_Real disjfactor2; /* (peak-capacity)/capacity * (large demands/nvars_t) */
   SCIP_Real cumfactor1;
   SCIP_Real resstrength1; /* overall strength */
   SCIP_Real resstrength2; /* timepoint wise maximum */

   /* helpful variables: */
   SCIP_Real globalpeak;
   SCIP_Real globalmaxdemand;

   /* get constraint data structure */
   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;
   capacity = consdata->capacity;
   globalpeak = 0.0;
   globalmaxdemand = 0.0;

   disjfactor2 = 0.0;
   cumfactor1 = 0.0;
   resstrength2 = 0.0;

   /* check each starting time (==each job, but inefficient) */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_Real peak;
      SCIP_Real maxdemand;
      SCIP_Real deltademand;
      int ndemands;
      int nlarge;

      int timepoint;
      int j;
      timepoint = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[v]));
      peak = consdata->demands[v];
      ndemands = 1;
      maxdemand = 0;
      nlarge = 0;

      if( consdata->demands[v] > capacity / 3 )
         nlarge++;

      for( j = 0; j < nvars; ++j )
      {
         int lb;

         if( j == v )
            continue;

         maxdemand = 0.0;
         lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[j]));

         if( lb <= timepoint && lb + consdata->durations[j] > timepoint )
         {
            peak += consdata->demands[j];
            ndemands++;

            if( consdata->demands[j] > consdata->capacity / 3 )
               nlarge++;
         }
      }

      deltademand = (SCIP_Real)peak / (SCIP_Real)ndemands;
      globalpeak = MAX(globalpeak, peak);
      globalmaxdemand = MAX(globalmaxdemand, maxdemand);

      if( peak > capacity )
      {
         disjfactor2 = MAX( disjfactor2, (peak-(SCIP_Real)capacity)/peak * (nlarge/(SCIP_Real)ndemands) );
         cumfactor1 = MAX( cumfactor1, (peak-capacity)/peak * (capacity-deltademand)/(SCIP_Real)capacity );
         resstrength2 = MAX(resstrength2, (capacity-maxdemand)/(peak-maxdemand) );
      }
   }

   resstrength1 = (capacity-globalmaxdemand) / (globalpeak-globalmaxdemand);

   consdata->maxpeak = SCIPconvertRealToInt(scip, globalpeak);
   consdata->disjfactor2 = disjfactor2;
   consdata->cumfactor1 = cumfactor1;
   consdata->resstrength2 = resstrength2;
   consdata->resstrength1 = resstrength1;

   /* get estimated res strength */
   {
      int* timepoints;
      SCIP_Real* estimateddemands;
      int ntimepoints;
      int maxdemand;
      SCIP_Real minfreecapacity;

      SCIP_CALL( SCIPallocBufferArray(scip, &timepoints, 2*nvars) );
      SCIP_CALL( SCIPallocBufferArray(scip, &estimateddemands, 2*nvars) );

      ntimepoints = 0;
      minfreecapacity = INT_MAX;


      SCIP_CALL( computeRelevantEnergyIntervals(scip, nvars, consdata->vars,
            consdata->durations, consdata->demands,
            capacity, consdata->hmin, consdata->hmax, &timepoints, &estimateddemands,
            &ntimepoints, &maxdemand, &minfreecapacity) );

      /* free buffer arrays */
      SCIPfreeBufferArray(scip, &estimateddemands);
      SCIPfreeBufferArray(scip, &timepoints);

      consdata->estimatedstrength = (SCIP_Real)(capacity - minfreecapacity) / (SCIP_Real) capacity;
   }

   SCIPstatisticPrintf("cumulative constraint<%s>: DISJ1=%g, DISJ2=%g, CUM=%g, RS1 = %g, RS2 = %g, EST = %g\n",
      SCIPconsGetName(cons), consdata->disjfactor1, disjfactor2, cumfactor1, resstrength1, resstrength2,
      consdata->estimatedstrength);

   return SCIP_OKAY;
}
#endif

/** gets the active variables together with the constant */
static
SCIP_RETCODE getActiveVar(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR**            var,                /**< pointer to store the active variable */
   int*                  scalar,             /**< pointer to store the scalar */
   int*                  constant            /**< pointer to store the constant */
   )
{
   if( !SCIPvarIsActive(*var) )
   {
      SCIP_Real realscalar;
      SCIP_Real realconstant;

      realscalar = 1.0;
      realconstant = 0.0;

      assert(SCIPvarGetStatus(*var) == SCIP_VARSTATUS_AGGREGATED);

      /* transform variable to active variable */
      SCIP_CALL( SCIPgetProbvarSum(scip, var, &realscalar, &realconstant) );
      assert(!SCIPisZero(scip, realscalar));
      assert(SCIPvarIsActive(*var));

      if( realconstant < 0.0 )
         (*constant) = -SCIPconvertRealToInt(scip, -realconstant);
      else
         (*constant) = SCIPconvertRealToInt(scip, realconstant);

      if( realscalar < 0.0 )
         (*scalar) = -SCIPconvertRealToInt(scip, -realscalar);
      else
         (*scalar) = SCIPconvertRealToInt(scip, realscalar);
   }
   else
   {
      (*scalar) = 1;
      (*constant) = 0;
   }

   assert(*scalar != 0);

   return SCIP_OKAY;
}

/** computes the total energy of all jobs */
static
int computeTotalEnergy(
   int*                  durations,          /**< array of job durations */
   int*                  demands,            /**< array of job demands */
   int                   njobs               /**< number of jobs */
   )
{
   int energy;
   int j;

   energy = 0;

   for( j = 0; j < njobs; ++j )
      energy += durations[j] * demands[j];

   return energy;
}

/**@} */

/**@name Default method to solve a cumulative condition
 *
 * @{
 */

/** solve single cumulative condition using SCIP and a single cumulative constraint */
static
SCIP_DECL_SOLVECUMULATIVE(solveCumulativeViaScipCp)
{
   SCIP* subscip;
   SCIP_VAR** subvars;
   SCIP_CONS* cons;
   SCIP_RETCODE retcode;
   char name[SCIP_MAXSTRLEN];
   int v;

   assert(njobs > 0);

   (*solved) = FALSE;
   (*infeasible) = FALSE;
   (*unbounded) = FALSE;
   (*error) = FALSE;

   SCIPdebugMessage("solve independent cumulative condition with %d variables\n", njobs);

   /* initialize the sub-problem */
   SCIP_CALL( SCIPcreate(&subscip) );

   /* copy all plugins */
   SCIP_CALL( SCIPincludeDefaultPlugins(subscip) );

   /* create the subproblem */
   SCIP_CALL( SCIPcreateProbBasic(subscip, "cumulative") );

   SCIP_CALL( SCIPallocMemoryArray(subscip, &subvars, njobs) );

   /* create for each job a start time variable */
   for( v = 0; v < njobs; ++v )
   {
      SCIP_Real objval;

      /* construct varibale name */
      (void) SCIPsnprintf(name, SCIP_MAXSTRLEN, "job%d", v);

      if( objvals == NULL )
         objval = 0.0;
      else
         objval = objvals[v];

      SCIP_CALL( SCIPcreateVarBasic(subscip, &subvars[v], name, ests[v], lsts[v], objval, SCIP_VARTYPE_INTEGER) );
      SCIP_CALL( SCIPaddVar(subscip, subvars[v]) );
   }

   /* create cumulative constraint */
   SCIP_CALL( SCIPcreateConsBasicCumulative(subscip, &cons, "cumulative",
         njobs, subvars, durations, demands, capacity) );

   /* set effective horizon */
   SCIP_CALL( SCIPsetHminCumulative(subscip, cons, hmin) );
   SCIP_CALL( SCIPsetHmaxCumulative(subscip, cons, hmax) );

   /* add cumulative constraint */
   SCIP_CALL( SCIPaddCons(subscip, cons) );
   SCIP_CALL( SCIPreleaseCons(subscip, &cons) );

   /* set CP solver settings
    *
    * @note This "meta" setting has to be set first since this call overwrite all parameters including for example the
    *       time limit.
    */
   SCIP_CALL( SCIPsetEmphasis(subscip, SCIP_PARAMEMPHASIS_CPSOLVER, TRUE) );

   /* do not abort subproblem on CTRL-C */
   SCIP_CALL( SCIPsetBoolParam(subscip, "misc/catchctrlc", FALSE) );

   /* disable output to console */
   SCIP_CALL( SCIPsetIntParam(subscip, "display/verblevel", 0) );

   /* set limits for the subproblem */
   SCIP_CALL( SCIPsetLongintParam(subscip, "limits/nodes", maxnodes) );
   SCIP_CALL( SCIPsetRealParam(subscip, "limits/time", timelimit) );
   SCIP_CALL( SCIPsetRealParam(subscip, "limits/memory", memorylimit) );

   /* forbid recursive call of heuristics and separators solving subMIPs
    * todo: really? This method was part of 3.0.1 but not in v31-Bugfix
    */
   SCIP_CALL( SCIPsetSubscipsOff(subscip, TRUE) );

   /* solve single cumulative constraint by branch and bound */
   retcode = SCIPsolve(subscip);

   if( retcode != SCIP_OKAY )
      (*error) = TRUE;
   else
   {
      SCIPdebugMessage("solved single cumulative condition with status %d\n", SCIPgetStatus(subscip));

      /* evaluated solution status */
      switch( SCIPgetStatus(subscip) )
      {
      case SCIP_STATUS_INFORUNBD:
      case SCIP_STATUS_INFEASIBLE:
         (*infeasible) = TRUE;
         (*solved) = TRUE;
         break;
      case SCIP_STATUS_UNBOUNDED:
         (*unbounded) = TRUE;
         (*solved) = TRUE;
         break;
      case SCIP_STATUS_OPTIMAL:
      {
         SCIP_SOL* sol;
         SCIP_Real solval;

         sol = SCIPgetBestSol(subscip);
         assert(sol != NULL);

         for( v = 0; v < njobs; ++v )
         {
            solval = SCIPgetSolVal(subscip, sol, subvars[v]);

            ests[v] = solval;
            lsts[v] = solval;
         }
         (*solved) = TRUE;
         break;
      }
      case SCIP_STATUS_NODELIMIT:
      case SCIP_STATUS_TOTALNODELIMIT:
      case SCIP_STATUS_TIMELIMIT:
      case SCIP_STATUS_MEMLIMIT:
      case SCIP_STATUS_USERINTERRUPT:
         /* transfer the global bound changes */
         for( v = 0; v < njobs; ++v )
         {
            ests[v] = SCIPvarGetLbGlobal(subvars[v]);
            lsts[v] = SCIPvarGetUbGlobal(subvars[v]);
         }
         (*solved) = FALSE;
         break;

      case SCIP_STATUS_UNKNOWN:
      case SCIP_STATUS_STALLNODELIMIT:
      case SCIP_STATUS_GAPLIMIT:
      case SCIP_STATUS_SOLLIMIT:
      case SCIP_STATUS_BESTSOLLIMIT:
      case SCIP_STATUS_RESTARTLIMIT:
         SCIPerrorMessage("invalid status code <%d>\n", SCIPgetStatus(subscip));
         return SCIP_INVALIDDATA;
      }
   }

   /* release all variables */
   for( v = 0; v < njobs; ++v )
   {
      SCIP_CALL( SCIPreleaseVar(subscip, &subvars[v]) );
   }

   SCIPfreeMemoryArray(subscip, &subvars);

   SCIP_CALL( SCIPfree(&subscip) );

   return SCIP_OKAY;
}

#if 0
/** solve single cumulative condition using SCIP and the time indexed formulation */
static
SCIP_DECL_SOLVECUMULATIVE(solveCumulativeViaScipMip)
{
   SCIP* subscip;
   SCIP_VAR*** binvars;
   SCIP_RETCODE retcode;
   char name[SCIP_MAXSTRLEN];
   int minest;
   int maxlct;
   int t;
   int v;

   assert(njobs > 0);

   (*solved) = FALSE;
   (*infeasible) = FALSE;
   (*unbounded) = FALSE;
   (*error) = FALSE;

   SCIPdebugMessage("solve independent cumulative condition with %d variables\n", njobs);

   /* initialize the sub-problem */
   SCIP_CALL( SCIPcreate(&subscip) );

   /* copy all plugins */
   SCIP_CALL( SCIPincludeDefaultPlugins(subscip) );

   /* create the subproblem */
   SCIP_CALL( SCIPcreateProbBasic(subscip, "cumulative") );

   SCIP_CALL( SCIPallocMemoryArray(subscip, &binvars, njobs) );

   minest = INT_MAX;
   maxlct = INT_MIN;

   /* create for each job and time step a binary variable which is one if this jobs starts at this time point and a set
    * partitioning constrain which forces that job starts
    */
   for( v = 0; v < njobs; ++v )
   {
      SCIP_CONS* cons;
      SCIP_Real objval;
      int timeinterval;
      int est;
      int lst;

      if( objvals == NULL )
         objval = 0.0;
      else
         objval = objvals[v];

      est = ests[v];
      lst = lsts[v];

      /* compute number of possible start points */
      timeinterval = lst - est + 1;
      assert(timeinterval > 0);

      /* compute the smallest earliest start time and largest latest completion time */
      minest = MIN(minest, est);
      maxlct = MAX(maxlct, lst + durations[v]);

      /* construct constraint name */
      (void) SCIPsnprintf(name, SCIP_MAXSTRLEN, "job_%d", v);

      SCIP_CALL( SCIPcreateConsBasicSetpart(subscip, &cons, name, 0, NULL) );

      SCIP_CALL( SCIPallocMemoryArray(subscip, &binvars[v], timeinterval) );

      for( t = 0; t < timeinterval; ++t )
      {
         SCIP_VAR* binvar;

         /* construct varibale name */
         (void) SCIPsnprintf(name, SCIP_MAXSTRLEN, "job_%d_time_%d", v, t + est);

         SCIP_CALL( SCIPcreateVarBasic(subscip, &binvar, name, 0.0, 1.0, objval, SCIP_VARTYPE_BINARY) );
         SCIP_CALL( SCIPaddVar(subscip, binvar) );

         /* add binary varibale to the set partitioning constraint which ensures that the job is started */
         SCIP_CALL( SCIPaddCoefSetppc(subscip, cons, binvar) );

         binvars[v][t] = binvar;
      }

      /* add and release the set partitioning constraint */
      SCIP_CALL( SCIPaddCons(subscip, cons) );
      SCIP_CALL( SCIPreleaseCons(subscip, &cons) );
   }

   /* adjusted the smallest earliest start time and the largest latest completion time with the effective horizon */
   hmin = MAX(hmin, minest);
   hmax = MIN(hmax, maxlct);
   assert(hmin > INT_MIN);
   assert(hmax < INT_MAX);
   assert(hmin < hmax);

   /* create for each time a knapsack constraint which ensures that the resource capacity is not exceeded */
   for( t = hmin; t < hmax; ++t )
   {
      SCIP_CONS* cons;

      /* construct constraint name */
      (void) SCIPsnprintf(name, SCIP_MAXSTRLEN, "time_%d", t);

      /* create an empty knapsack constraint */
      SCIP_CALL( SCIPcreateConsBasicKnapsack(subscip, &cons, name, 0, NULL, NULL, (SCIP_Longint)capacity) );

      /* add all jobs which potentially can be processed at that time point */
      for( v = 0; v < njobs; ++v )
      {
         int duration;
         int demand;
         int start;
         int end;
         int est;
         int lst;
         int k;

         est = ests[v];
         lst = lsts[v] ;

         duration = durations[v];
         assert(duration > 0);

         /* check if the varibale is processed potentially at time point t */
         if( t < est || t >= lst + duration )
            continue;

         demand = demands[v];
         assert(demand >= 0);

         start = MAX(t - duration + 1, est);
         end = MIN(t, lst);

         assert(start <= end);

         for( k = start; k <= end; ++k )
         {
            assert(binvars[v][k] != NULL);
            SCIP_CALL( SCIPaddCoefKnapsack(subscip, cons, binvars[v][k], (SCIP_Longint) demand) );
         }
      }

      /* add and release the knapsack constraint */
      SCIP_CALL( SCIPaddCons(subscip, cons) );
      SCIP_CALL( SCIPreleaseCons(subscip, &cons) );
   }

   /* do not abort subproblem on CTRL-C */
   SCIP_CALL( SCIPsetBoolParam(subscip, "misc/catchctrlc", FALSE) );

   /* disable output to console */
   SCIP_CALL( SCIPsetIntParam(subscip, "display/verblevel", 0) );

   /* set limits for the subproblem */
   SCIP_CALL( SCIPsetLongintParam(subscip, "limits/nodes", maxnodes) );
   SCIP_CALL( SCIPsetRealParam(subscip, "limits/time", timelimit) );
   SCIP_CALL( SCIPsetRealParam(subscip, "limits/memory", memorylimit) );

   /* solve single cumulative constraint by branch and bound */
   retcode = SCIPsolve(subscip);

   if( retcode != SCIP_OKAY )
      (*error) = TRUE;
   else
   {
      SCIPdebugMessage("solved single cumulative condition with status %d\n", SCIPgetStatus(subscip));

      /* evaluated solution status */
      switch( SCIPgetStatus(subscip) )
      {
      case SCIP_STATUS_INFORUNBD:
      case SCIP_STATUS_INFEASIBLE:
         (*infeasible) = TRUE;
         (*solved) = TRUE;
         break;
      case SCIP_STATUS_UNBOUNDED:
         (*unbounded) = TRUE;
         (*solved) = TRUE;
         break;
      case SCIP_STATUS_OPTIMAL:
      {
         SCIP_SOL* sol;

         sol = SCIPgetBestSol(subscip);
         assert(sol != NULL);

         for( v = 0; v < njobs; ++v )
         {
            int timeinterval;
            int est;
            int lst;

            est = ests[v];
            lst = lsts[v];

            /* compute number of possible start points */
            timeinterval = lst - est + 1;

            /* check which binary varibale is set to one */
            for( t = 0; t < timeinterval; ++t )
            {
               if( SCIPgetSolVal(subscip, sol, binvars[v][t]) > 0.5 )
               {
                  ests[v] = est + t;
                  lsts[v] = est + t;
                  break;
               }
            }
         }

         (*solved) = TRUE;
         break;
      }
      case SCIP_STATUS_NODELIMIT:
      case SCIP_STATUS_TOTALNODELIMIT:
      case SCIP_STATUS_TIMELIMIT:
      case SCIP_STATUS_MEMLIMIT:
      case SCIP_STATUS_USERINTERRUPT:
         /* transfer the global bound changes */
         for( v = 0; v < njobs; ++v )
         {
            int timeinterval;
            int est;
            int lst;

            est = ests[v];
            lst = lsts[v];

            /* compute number of possible start points */
            timeinterval = lst - est + 1;

            /* check which binary varibale is the first binary varibale which is not globally fixed to zero */
            for( t = 0; t < timeinterval; ++t )
            {
               if( SCIPvarGetUbGlobal(binvars[v][t]) > 0.5 )
               {
                  ests[v] = est + t;
                  break;
               }
            }

            /* check which binary varibale is the last binary varibale which is not globally fixed to zero */
            for( t = timeinterval - 1; t >= 0; --t )
            {
               if( SCIPvarGetUbGlobal(binvars[v][t]) > 0.5 )
               {
                  lsts[v] = est + t;
                  break;
               }
            }
         }
         (*solved) = FALSE;
         break;

      case SCIP_STATUS_UNKNOWN:
      case SCIP_STATUS_STALLNODELIMIT:
      case SCIP_STATUS_GAPLIMIT:
      case SCIP_STATUS_SOLLIMIT:
      case SCIP_STATUS_BESTSOLLIMIT:
         SCIPerrorMessage("invalid status code <%d>\n", SCIPgetStatus(subscip));
         return SCIP_INVALIDDATA;
      }
   }

   /* release all variables */
   for( v = 0; v < njobs; ++v )
   {
      int timeinterval;
      int est;
      int lst;

      est = ests[v];
      lst = lsts[v];

      /* compute number of possible start points */
      timeinterval = lst - est + 1;

      for( t = 0; t < timeinterval; ++t )
      {
         SCIP_CALL( SCIPreleaseVar(subscip, &binvars[v][t]) );
      }
      SCIPfreeMemoryArray(subscip, &binvars[v]);
   }

   SCIPfreeMemoryArray(subscip, &binvars);

   SCIP_CALL( SCIPfree(&subscip) );

   return SCIP_OKAY;
}
#endif

/**@} */

/**@name Constraint handler data
 *
 * Method used to create and free the constraint handler data when including and removing the cumulative constraint
 * handler.
 *
 * @{
 */

/** creates constaint handler data for cumulative constraint handler */
static
SCIP_RETCODE conshdlrdataCreate(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA**   conshdlrdata,       /**< pointer to store the constraint handler data */
   SCIP_EVENTHDLR*       eventhdlr           /**< event handler */
   )
{
   /* create precedence constraint handler data */
   assert(scip != NULL);
   assert(conshdlrdata != NULL);
   assert(eventhdlr != NULL);

   SCIP_CALL( SCIPallocMemory(scip, conshdlrdata) );

   /* set event handler for checking if bounds of start time variables are tighten */
   (*conshdlrdata)->eventhdlr = eventhdlr;

   /* set default methed for solving single cumulative conditions using SCIP and a CP model */
   (*conshdlrdata)->solveCumulative = solveCumulativeViaScipCp;

#ifdef SCIP_STATISTIC
   (*conshdlrdata)->nlbtimetable = 0;
   (*conshdlrdata)->nubtimetable = 0;
   (*conshdlrdata)->ncutofftimetable = 0;
   (*conshdlrdata)->nlbedgefinder = 0;
   (*conshdlrdata)->nubedgefinder = 0;
   (*conshdlrdata)->ncutoffedgefinder = 0;
   (*conshdlrdata)->ncutoffoverload = 0;
   (*conshdlrdata)->ncutoffoverloadTTEF = 0;

   (*conshdlrdata)->nirrelevantjobs = 0;
   (*conshdlrdata)->nalwaysruns = 0;
   (*conshdlrdata)->nremovedlocks = 0;
   (*conshdlrdata)->ndualfixs = 0;
   (*conshdlrdata)->ndecomps = 0;
   (*conshdlrdata)->ndualbranchs = 0;
   (*conshdlrdata)->nallconsdualfixs = 0;
   (*conshdlrdata)->naddedvarbounds = 0;
   (*conshdlrdata)->naddeddisjunctives = 0;
#endif

   return SCIP_OKAY;
}

/** frees constraint handler data for logic or constraint handler */
static
void conshdlrdataFree(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA**   conshdlrdata        /**< pointer to the constraint handler data */
   )
{
   assert(conshdlrdata != NULL);
   assert(*conshdlrdata != NULL);

   SCIPfreeMemory(scip, conshdlrdata);
}

/**@} */


/**@name Constraint data methods
 *
 * @{
 */

/** catches bound change events for all variables in transformed cumulative constraint */
static
SCIP_RETCODE consdataCatchEvents(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< cumulative constraint data */
   SCIP_EVENTHDLR*       eventhdlr           /**< event handler to call for the event processing */
   )
{
   int v;

   assert(scip != NULL);
   assert(consdata != NULL);
   assert(eventhdlr != NULL);

   /* catch event for every single variable */
   for( v = 0; v < consdata->nvars; ++v )
   {
      SCIP_CALL( SCIPcatchVarEvent(scip, consdata->vars[v],
            SCIP_EVENTTYPE_BOUNDTIGHTENED, eventhdlr, (SCIP_EVENTDATA*)consdata, NULL) );
   }

   return SCIP_OKAY;
}

/** drops events for variable at given position */
static
SCIP_RETCODE consdataDropEvents(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< cumulative constraint data */
   SCIP_EVENTHDLR*       eventhdlr,          /**< event handler to call for the event processing */
   int                   pos                 /**< array position of variable to catch bound change events for */
   )
{
   assert(scip != NULL);
   assert(consdata != NULL);
   assert(eventhdlr != NULL);
   assert(0 <= pos && pos < consdata->nvars);
   assert(consdata->vars[pos] != NULL);

   SCIP_CALL( SCIPdropVarEvent(scip, consdata->vars[pos],
         SCIP_EVENTTYPE_BOUNDTIGHTENED, eventhdlr, (SCIP_EVENTDATA*)consdata, -1) );

   return SCIP_OKAY;
}

/** drops bound change events for all variables in transformed linear constraint */
static
SCIP_RETCODE consdataDropAllEvents(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< linear constraint data */
   SCIP_EVENTHDLR*       eventhdlr           /**< event handler to call for the event processing */
   )
{
   int v;

   assert(scip != NULL);
   assert(consdata != NULL);

   /* drop event of every single variable */
   for( v = 0; v < consdata->nvars; ++v )
   {
      SCIP_CALL( consdataDropEvents(scip, consdata, eventhdlr, v) );
   }

   return SCIP_OKAY;
}

/** initialize variable lock data structure */
static
void initializeLocks(
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   SCIP_Bool             locked              /**< should the variable be locked? */
   )
{
   int nvars;
   int v;

   nvars = consdata->nvars;

   /* initialize locking arrays */
   for( v = 0; v < nvars; ++v )
   {
      consdata->downlocks[v] = locked;
      consdata->uplocks[v] = locked;
   }
}

/** creates constraint data of cumulative constraint */
static
SCIP_RETCODE consdataCreate(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA**       consdata,           /**< pointer to consdata */
   SCIP_VAR**            vars,               /**< array of integer variables */
   SCIP_CONS**           linkingconss,       /**< array of linking constraints for the integer variables, or NULL */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   nvars,              /**< number of variables */
   int                   capacity,           /**< available cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool             check               /**< is the corresponding constraint a check constraint */
   )
{
   int v;

   assert(scip != NULL);
   assert(consdata != NULL);
   assert(vars != NULL || nvars > 0);
   assert(demands != NULL);
   assert(durations != NULL);
   assert(capacity >= 0);
   assert(hmin >= 0);
   assert(hmin < hmax);

   /* create constraint data */
   SCIP_CALL( SCIPallocBlockMemory(scip, consdata) );

   (*consdata)->hmin = hmin;
   (*consdata)->hmax = hmax;

   (*consdata)->capacity = capacity;
   (*consdata)->demandrows = NULL;
   (*consdata)->demandrowssize = 0;
   (*consdata)->ndemandrows = 0;
   (*consdata)->scoverrows = NULL;
   (*consdata)->nscoverrows = 0;
   (*consdata)->scoverrowssize = 0;
   (*consdata)->bcoverrows = NULL;
   (*consdata)->nbcoverrows = 0;
   (*consdata)->bcoverrowssize = 0;
   (*consdata)->nvars = nvars;
   (*consdata)->varssize = nvars;
   (*consdata)->signature = 0;
   (*consdata)->validsignature = FALSE;
   (*consdata)->normalized = FALSE;
   (*consdata)->covercuts = FALSE;
   (*consdata)->propagated = FALSE;
   (*consdata)->varbounds = FALSE;
   (*consdata)->triedsolving = FALSE;

   if( nvars > 0 )
   {
      assert(vars != NULL); /* for flexelint */

      SCIP_CALL( SCIPduplicateBlockMemoryArray(scip, &(*consdata)->vars, vars, nvars) );
      SCIP_CALL( SCIPduplicateBlockMemoryArray(scip, &(*consdata)->demands, demands, nvars) );
      SCIP_CALL( SCIPduplicateBlockMemoryArray(scip, &(*consdata)->durations, durations, nvars) );
      (*consdata)->linkingconss = NULL;

      SCIP_CALL( SCIPallocBlockMemoryArray(scip, &(*consdata)->downlocks, nvars) );
      SCIP_CALL( SCIPallocBlockMemoryArray(scip, &(*consdata)->uplocks, nvars) );

      /* initialize variable lock data structure; the locks are only used if the contraint is a check constraint */
      initializeLocks(*consdata, check);

      if( linkingconss != NULL )
      {
         SCIP_CALL( SCIPduplicateBlockMemoryArray(scip, &(*consdata)->linkingconss, linkingconss, nvars) );
      }

      /* transform variables, if they are not yet transformed */
      if( SCIPisTransformed(scip) )
      {
         SCIPdebugMessage("get tranformed variables and constraints\n");

         /* get transformed variables and do NOT captures these */
         SCIP_CALL( SCIPgetTransformedVars(scip, (*consdata)->nvars, (*consdata)->vars, (*consdata)->vars) );

         /* multi-aggregated variables cannot be replaced by active variable; therefore we mark all variables for not
          * been multi-aggregated
          */
         for( v = 0; v < nvars; ++v )
         {
            SCIP_CALL( SCIPmarkDoNotMultaggrVar(scip, (*consdata)->vars[v]) );
         }

         if( linkingconss != NULL )
         {
            /* get transformed constraints and captures these */
            SCIP_CALL( SCIPtransformConss(scip, (*consdata)->nvars, (*consdata)->linkingconss, (*consdata)->linkingconss) );

            for( v = 0; v < nvars; ++v )
               assert(SCIPgetConsLinking(scip, (*consdata)->vars[v]) == (*consdata)->linkingconss[v]);
         }
      }
   }
   else
   {
      (*consdata)->vars = NULL;
      (*consdata)->downlocks = NULL;
      (*consdata)->uplocks = NULL;
      (*consdata)->demands = NULL;
      (*consdata)->durations = NULL;
      (*consdata)->linkingconss = NULL;
   }

   /* initialize values for running propagation algorithms efficiently */
   (*consdata)->resstrength1 = -1.0;
   (*consdata)->resstrength2 = -1.0;
   (*consdata)->cumfactor1 = -1.0;
   (*consdata)->disjfactor1 = -1.0;
   (*consdata)->disjfactor2 = -1.0;
   (*consdata)->estimatedstrength = -1.0;

   SCIPstatistic( (*consdata)->maxpeak = -1 );

   return SCIP_OKAY;
}

/** releases LP rows of constraint data and frees rows array */
static
SCIP_RETCODE consdataFreeRows(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA**       consdata            /**< constraint data */
   )
{
   int r;

   assert(consdata != NULL);
   assert(*consdata != NULL);

   for( r = 0; r < (*consdata)->ndemandrows; ++r )
   {
      assert((*consdata)->demandrows[r] != NULL);
      SCIP_CALL( SCIPreleaseRow(scip, &(*consdata)->demandrows[r]) );
   }

   SCIPfreeBlockMemoryArrayNull(scip, &(*consdata)->demandrows, (*consdata)->demandrowssize);

   (*consdata)->ndemandrows = 0;
   (*consdata)->demandrowssize = 0;

   /* free rows of cover cuts */
   for( r = 0; r < (*consdata)->nscoverrows; ++r )
   {
      assert((*consdata)->scoverrows[r] != NULL);
      SCIP_CALL( SCIPreleaseRow(scip, &(*consdata)->scoverrows[r]) );
   }

   SCIPfreeBlockMemoryArrayNull(scip, &(*consdata)->scoverrows, (*consdata)->scoverrowssize);

   (*consdata)->nscoverrows = 0;
   (*consdata)->scoverrowssize = 0;

   for( r = 0; r < (*consdata)->nbcoverrows; ++r )
   {
      assert((*consdata)->bcoverrows[r] != NULL);
      SCIP_CALL( SCIPreleaseRow(scip, &(*consdata)->bcoverrows[r]) );
   }

   SCIPfreeBlockMemoryArrayNull(scip, &(*consdata)->bcoverrows, (*consdata)->bcoverrowssize);

   (*consdata)->nbcoverrows = 0;
   (*consdata)->bcoverrowssize = 0;

   (*consdata)->covercuts = FALSE;

   return SCIP_OKAY;
}

/** frees a cumulative constraint data */
static
SCIP_RETCODE consdataFree(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA**       consdata            /**< pointer to linear constraint data */
   )
{
   int varssize;
   int nvars;

   assert(consdata != NULL);
   assert(*consdata != NULL);

   nvars =  (*consdata)->nvars;
   varssize = (*consdata)->varssize;

   if( varssize > 0 )
   {
      int v;

      /* release and free the rows */
      SCIP_CALL( consdataFreeRows(scip, consdata) );

      /* release the linking constraints if they were generated */
      if( (*consdata)->linkingconss != NULL )
      {
         for( v = nvars-1; v >= 0; --v )
         {
            assert((*consdata)->linkingconss[v] != NULL );
            SCIP_CALL( SCIPreleaseCons(scip, &(*consdata)->linkingconss[v]) );
         }

         SCIPfreeBlockMemoryArray(scip, &(*consdata)->linkingconss, varssize);
      }

      /* free arrays */
      SCIPfreeBlockMemoryArray(scip, &(*consdata)->downlocks, varssize);
      SCIPfreeBlockMemoryArray(scip, &(*consdata)->uplocks, varssize);
      SCIPfreeBlockMemoryArray(scip, &(*consdata)->durations, varssize);
      SCIPfreeBlockMemoryArray(scip, &(*consdata)->demands, varssize);
      SCIPfreeBlockMemoryArray(scip, &(*consdata)->vars, varssize);
   }

   /* free memory */
   SCIPfreeBlockMemory(scip, consdata);

   return SCIP_OKAY;
}

/** prints cumulative constraint to file stream */
static
void consdataPrint(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< cumulative constraint data */
   FILE*                 file                /**< output file (or NULL for standard output) */
   )
{
   int v;

   assert(consdata != NULL);

   /* print coefficients */
   SCIPinfoMessage( scip, file, "cumulative(");

   for( v = 0; v < consdata->nvars; ++v )
   {
      assert(consdata->vars[v] != NULL);
      if( v > 0 )
         SCIPinfoMessage(scip, file, ", ");
      SCIPinfoMessage(scip, file, "<%s>[%g,%g](%d)[%d]", SCIPvarGetName(consdata->vars[v]),
         SCIPvarGetLbGlobal(consdata->vars[v]), SCIPvarGetUbGlobal(consdata->vars[v]),
         consdata->durations[v], consdata->demands[v]);
   }
   SCIPinfoMessage(scip, file, ")[%d,%d) <= %d", consdata->hmin, consdata->hmax, consdata->capacity);
}

/** deletes coefficient at given position from constraint data */
static
SCIP_RETCODE consdataDeletePos(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< cumulative constraint data */
   SCIP_CONS*            cons,               /**< knapsack constraint */
   int                   pos                 /**< position of coefficient to delete */
   )
{
   SCIP_CONSHDLR* conshdlr;
   SCIP_CONSHDLRDATA* conshdlrdata;

   assert(scip != NULL);
   assert(consdata != NULL);
   assert(cons != NULL);
   assert(SCIPconsIsTransformed(cons));
   assert(!SCIPinProbing(scip));

   SCIPdebugMessage("cumulative constraint <%s>: remove variable <%s>\n",
      SCIPconsGetName(cons), SCIPvarGetName(consdata->vars[pos]));

   /* remove the rounding locks for the deleted variable */
   SCIP_CALL( SCIPunlockVarCons(scip, consdata->vars[pos], cons, consdata->downlocks[pos], consdata->uplocks[pos]) );

   consdata->downlocks[pos] = FALSE;
   consdata->uplocks[pos] = FALSE;

   if( consdata->linkingconss != NULL )
   {
      SCIP_CALL( SCIPreleaseCons(scip, &consdata->linkingconss[pos]) );
   }

   /* get event handler */
   conshdlr = SCIPconsGetHdlr(cons);
   assert(conshdlr != NULL);
   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);
   assert(conshdlrdata->eventhdlr != NULL);

   /* drop events */
   SCIP_CALL( consdataDropEvents(scip, consdata, conshdlrdata->eventhdlr, pos) );

   SCIPdebugMessage("remove variable <%s>[%g,%g] from cumulative constraint <%s>\n",
      SCIPvarGetName(consdata->vars[pos]), SCIPvarGetLbGlobal(consdata->vars[pos]), SCIPvarGetUbGlobal(consdata->vars[pos]), SCIPconsGetName(cons));


   /* in case the we did not remove the variable in the last slot of the arrays we move the current last to this
    * position
    */
   if( pos != consdata->nvars - 1 )
   {
      consdata->vars[pos] = consdata->vars[consdata->nvars-1];
      consdata->downlocks[pos] = consdata->downlocks[consdata->nvars-1];
      consdata->uplocks[pos] = consdata->uplocks[consdata->nvars-1];
      consdata->demands[pos] = consdata->demands[consdata->nvars-1];
      consdata->durations[pos] = consdata->durations[consdata->nvars-1];

      if( consdata->linkingconss != NULL )
      {
         consdata->linkingconss[pos]= consdata->linkingconss[consdata->nvars-1];
      }
   }

   consdata->nvars--;
   consdata->validsignature = FALSE;
   consdata->normalized = FALSE;

   return SCIP_OKAY;
}

/** collect linking constraints for each integer variable */
static
SCIP_RETCODE consdataCollectLinkingCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata            /**< pointer to consdata */
   )
{
   int nvars;
   int v;

   assert(scip != NULL);
   assert(consdata != NULL);

   nvars = consdata->nvars;
   assert(nvars > 0);
   assert(consdata->linkingconss == NULL);

   SCIP_CALL( SCIPallocBlockMemoryArray(scip, &consdata->linkingconss, consdata->varssize) );

   for( v = 0; v < nvars; ++v )
   {
      SCIP_CONS* cons;
      SCIP_VAR* var;

      var = consdata->vars[v];
      assert(var != NULL);

      SCIPdebugMessage("linking constraint (%d of %d) for variable <%s>\n", v+1, nvars, SCIPvarGetName(var));

      /* create linking constraint if it does not exist yet */
      if( !SCIPexistsConsLinking(scip, var) )
      {
         char name[SCIP_MAXSTRLEN];

         (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "link(%s)", SCIPvarGetName(var));

         /* creates and captures an linking constraint */
         SCIP_CALL( SCIPcreateConsLinking(scip, &cons, name, var, NULL, 0, 0,
               TRUE, TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, FALSE, FALSE /*TRUE*/, FALSE) );
         SCIP_CALL( SCIPaddCons(scip, cons) );
         consdata->linkingconss[v] = cons;
      }
      else
      {
         consdata->linkingconss[v] = SCIPgetConsLinking(scip, var);
         SCIP_CALL( SCIPcaptureCons(scip, consdata->linkingconss[v]) );
      }

      assert(SCIPexistsConsLinking(scip, var));
      assert(consdata->linkingconss[v] != NULL);
      assert(strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(consdata->linkingconss[v])), "linking") == 0 );
      assert(SCIPgetConsLinking(scip, var) == consdata->linkingconss[v]);
   }

   return SCIP_OKAY;
}

/**@} */


/**@name Check methods
 *
 * @{
 */

/** check for the given starting time variables with their demands and durations if the cumulative conditions for the
 *  given solution is satisfied
 */
static
SCIP_RETCODE checkCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_SOL*             sol,                /**< primal solution, or NULL for current LP/pseudo solution */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            violated,           /**< pointer to store if the cumulative condition is violated */
   SCIP_CONS*            cons,               /**< constraint which is checked */
   SCIP_Bool             printreason         /**< should the reason for the violation be printed? */
   )
{
   int* startsolvalues;       /* stores when each job is starting */
   int* endsolvalues;         /* stores when each job ends */
   int* startindices;         /* we will sort the startsolvalues, thus we need to know which index of a job it corresponds to */
   int* endindices;           /* we will sort the endsolvalues, thus we need to know which index of a job it corresponds to */

   int freecapacity;
   int curtime;            /* point in time which we are just checking */
   int endindex;           /* index of endsolvalues with: endsolvalues[endindex] > curtime */
   int j;

   assert(scip != NULL);
   assert(violated != NULL);

   (*violated) = FALSE;

   if( nvars == 0 )
      return SCIP_OKAY;

   assert(vars != NULL);
   assert(demands != NULL);
   assert(durations != NULL);

   /* compute time points where we have to check whether capacity constraint is infeasible or not */
   SCIP_CALL( SCIPallocBufferArray(scip, &startsolvalues, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endsolvalues, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   /* assign variables, start and endpoints to arrays */
   for ( j = 0; j < nvars; ++j )
   {
      int solvalue;

      /* the constraint of the cumulative constraint handler should be called after the integrality check */
      assert(SCIPisFeasIntegral(scip, SCIPgetSolVal(scip, sol, vars[j])));

      solvalue = SCIPconvertRealToInt(scip, SCIPgetSolVal(scip, sol, vars[j]));

      /* we need to ensure that we check at least one time point during the effective horizon; therefore we project all
       * jobs which start before hmin to hmin
       */
      startsolvalues[j] = MAX(solvalue, hmin);
      startindices[j] = j;

      endsolvalues[j] = MAX(solvalue + durations[j], hmin);
      endindices[j] = j;
   }

   /* sort the arrays not-decreasing according to start solution values and end solution values (and sort the
    * corresponding indices in the same way)
    */
   SCIPsortIntInt(startsolvalues, startindices, nvars);
   SCIPsortIntInt(endsolvalues, endindices, nvars);

   endindex = 0;
   freecapacity = capacity;

   /* check each start point of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      /* only check intervals [hmin,hmax) */
      curtime = startsolvalues[j];

      if( curtime >= hmax )
         break;

      /* subtract all capacity needed up to this point */
      freecapacity -= demands[startindices[j]];
      while( j+1 < nvars && startsolvalues[j+1] == curtime )
      {
         j++;
         freecapacity -= demands[startindices[j]];
      }

      /* free all capacity usages of jobs that are no longer running */
      while( endindex < nvars && curtime >= endsolvalues[endindex] )
      {
         freecapacity += demands[endindices[endindex]];
         ++endindex;
      }
      assert(freecapacity <= capacity);

      /* check freecapacity to be smaller than zero */
      if( freecapacity < 0 && curtime >= hmin )
      {
         SCIPdebugMessage("freecapacity = %3d\n", freecapacity);
         (*violated) = TRUE;

         if( printreason )
         {
            int i;

            /* first state the violated constraints */
            SCIP_CALL( SCIPprintCons(scip, cons, NULL) );

            /* second state the reason */
            SCIPinfoMessage(scip, NULL,
               ";\nviolation: at time point %d available capacity = %d, needed capacity = %d\n",
               curtime, capacity, capacity - freecapacity);

            for( i = 0; i <= j; ++i )
            {
               if( startsolvalues[i] + durations[startindices[i]] > curtime )
               {
                  SCIPinfoMessage(scip, NULL, "activity %s, start = %i, duration = %d, demand = %d \n",
                     SCIPvarGetName(vars[startindices[i]]), startsolvalues[i], durations[startindices[i]],
                     demands[startindices[i]]);
               }
            }
         }
         break;
      }
   } /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endsolvalues);
   SCIPfreeBufferArray(scip, &startsolvalues);

   return SCIP_OKAY;
}

/** check if the given constrait is valid; checks each starting point of a job whether the remaining capacity is at
 *  least zero or not. If not (*violated) is set to TRUE
 */
static
SCIP_RETCODE checkCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   SCIP_SOL*             sol,                /**< primal solution, or NULL for current LP/pseudo solution */
   SCIP_Bool*            violated,           /**< pointer to store if the constraint is violated */
   SCIP_Bool             printreason         /**< should the reason for the violation be printed? */
   )
{
   SCIP_CONSDATA* consdata;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(violated != NULL);

   SCIPdebugMessage("check cumulative constraints <%s>\n", SCIPconsGetName(cons));

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   /* check the cumulative condition */
   SCIP_CALL( checkCumulativeCondition(scip, sol, consdata->nvars, consdata->vars,
         consdata->durations, consdata->demands, consdata->capacity, consdata->hmin, consdata->hmax,
         violated, cons, printreason) );

   return SCIP_OKAY;
}

/**@} */

/**@name Conflict analysis
 *
 * @{
 */

/** resolves the propagation of the core time algorithm */
static
SCIP_RETCODE resolvePropagationCoretimes(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             infervar,           /**< inference variable */
   int                   inferdemand,        /**< demand of the inference variable */
   int                   inferpeak,          /**< time point which causes the propagation */
   int                   relaxedpeak,        /**< relaxed time point which would be sufficient to be proved */
   SCIP_BDCHGIDX*        bdchgidx,           /**< the index of the bound change, representing the point of time where the change took place */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   int*                  provedpeak,         /**< pointer to store the actually proved peak, or NULL */
   SCIP_Bool*            explanation         /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   )
{
   SCIP_VAR* var;
   SCIP_Bool* reported;
   int duration;
   int maxlst;
   int minect;
   int ect;
   int lst;
   int j;

   assert(SCIPgetStage(scip) == SCIP_STAGE_SOLVING || SCIPinProbing(scip));

   SCIPdebugMessage("variable <%s>: (demand %d) resolve propagation of core time algorithm (peak %d)\n",
      SCIPvarGetName(infervar), inferdemand, inferpeak);
   assert(nvars > 0);

   /* adjusted capacity */
   capacity -= inferdemand;
   maxlst = INT_MIN;
   minect = INT_MAX;

   SCIP_CALL( SCIPallocBufferArray(scip, &reported, nvars) );
   BMSclearMemoryArray(reported, nvars);

   /* first we loop over all variables and adjust the capacity with those jobs which provide a global core at the
    * inference peak and those where the current conflict bounds provide a core at the inference peak
    */
   for( j = 0; j < nvars && capacity >= 0; ++j )
   {
      var = vars[j];
      assert(var != NULL);

      /* skip inference variable */
      if( var == infervar )
         continue;

      duration = durations[j];
      assert(duration > 0);

      /* compute cores of jobs; if core overlaps interval of inference variable add this job to the array */
      assert(!SCIPvarIsActive(var) || SCIPisFeasEQ(scip, SCIPvarGetUbAtIndex(var, bdchgidx, TRUE), SCIPvarGetUbAtIndex(var, bdchgidx, FALSE)));
      assert(SCIPisFeasIntegral(scip, SCIPvarGetUbAtIndex(var, bdchgidx, TRUE)));
      assert(!SCIPvarIsActive(var) || SCIPisFeasEQ(scip, SCIPvarGetLbAtIndex(var, bdchgidx, TRUE), SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)));
      assert(SCIPisFeasIntegral(scip, SCIPvarGetLbAtIndex(var, bdchgidx, TRUE)));

      SCIPdebugMessage("variable <%s>: glb=[%g,%g] conflict=[%g,%g] (duration %d, demand %d)\n",
         SCIPvarGetName(var),  SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var),
         SCIPgetConflictVarLb(scip, var), SCIPgetConflictVarUb(scip, var), duration, demands[j]);

      ect = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var)) + duration;
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));

      /* check if the inference peak is part of the global bound core; if so we decreasing the capacity by the demand of
       * that job without adding it the explanation
       */
      if( inferpeak < ect && lst <= inferpeak )
      {
         capacity -= demands[j];
         reported[j] = TRUE;

         maxlst = MAX(maxlst, lst);
         minect = MIN(minect, ect);
         assert(maxlst < minect);

         if( explanation != NULL )
            explanation[j] = TRUE;

         continue;
      }

      /* collect the conflict bound core (the conflict bounds are those bounds which are already part of the conflict)
       * hence these bound are already reported by other resolve propation steps. In case a bound (lower or upper) is
       * not part of the conflict yet we get the global bounds back.
       */
      ect = SCIPconvertRealToInt(scip, SCIPgetConflictVarLb(scip, var)) + duration;
      lst = SCIPconvertRealToInt(scip, SCIPgetConflictVarUb(scip, var));

      /* check if the inference peak is part of the conflict bound core; if so we decreasing the capacity by the demand
       * of that job without and collect the job as part of the explanation
       *
       * @note we do not need to reported that job to SCIP since the required bounds are already reported
       */
      if( inferpeak < ect && lst <= inferpeak )
      {
         capacity -= demands[j];
         reported[j] = TRUE;

         maxlst = MAX(maxlst, lst);
         minect = MIN(minect, ect);
         assert(maxlst < minect);

         if( explanation != NULL )
            explanation[j] = TRUE;
      }
   }

   if( capacity >= 0 )
   {
      int* cands;
      int* canddemands;
      int ncands;
      int c;

      SCIP_CALL( SCIPallocBufferArray(scip, &cands, nvars) );
      SCIP_CALL( SCIPallocBufferArray(scip, &canddemands, nvars) );
      ncands = 0;

      /* collect all cores of the variables which lay in the considered time window except the inference variable */
      for( j = 0; j < nvars; ++j )
      {
         var = vars[j];
         assert(var != NULL);

         /* skip inference variable */
         if( var == infervar || reported[j] )
            continue;

         duration = durations[j];
         assert(duration > 0);

         /* compute cores of jobs; if core overlaps interval of inference variable add this job to the array */
         assert(!SCIPvarIsActive(var) || SCIPisFeasEQ(scip, SCIPvarGetUbAtIndex(var, bdchgidx, TRUE), SCIPvarGetUbAtIndex(var, bdchgidx, FALSE)));
         assert(SCIPisFeasIntegral(scip, SCIPvarGetUbAtIndex(var, bdchgidx, TRUE)));
         assert(!SCIPvarIsActive(var) || SCIPisFeasEQ(scip, SCIPvarGetLbAtIndex(var, bdchgidx, TRUE), SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)));
         assert(SCIPisFeasIntegral(scip, SCIPvarGetLbAtIndex(var, bdchgidx, TRUE)));

         /* collect local core information */
         ect = SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)) + duration;
         lst = SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE));

         SCIPdebugMessage("variable <%s>: loc=[%g,%g] glb=[%g,%g] (duration %d, demand %d)\n",
            SCIPvarGetName(var), SCIPvarGetLbAtIndex(var, bdchgidx, FALSE), SCIPvarGetUbAtIndex(var, bdchgidx, FALSE),
            SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration, demands[j]);

         /* check if the inference peak is part of the core */
         if( inferpeak < ect && lst <= inferpeak )
         {
            cands[ncands] = j;
            canddemands[ncands] = demands[j];
            ncands++;

            capacity -= demands[j];
         }
      }

      /* sort candidates indices w.r.t. their demands */
      SCIPsortDownIntInt(canddemands, cands, ncands);

      assert(capacity < 0);
      assert(ncands > 0);

      /* greedily remove candidates form the list such that the needed capacity is still exceeded */
      while( capacity + canddemands[ncands-1] < 0 )
      {
         ncands--;
         capacity += canddemands[ncands];
         assert(ncands > 0);
      }

      /* compute the size (number of time steps) of the job cores */
      for( c = 0; c < ncands; ++c )
      {
         var = vars[cands[c]];
         assert(var != NULL);

         duration = durations[cands[c]];

         ect = SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)) + duration;
         lst = SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE));

         maxlst = MAX(maxlst, lst);
         minect = MIN(minect, ect);
         assert(maxlst < minect);
      }

      SCIPdebugMessage("infer peak %d, relaxed peak %d, lst %d, ect %d\n", inferpeak, relaxedpeak, maxlst, minect);
      assert(inferpeak >= maxlst);
      assert(inferpeak < minect);

      /* check if the collect variable are sufficient to prove the relaxed bound (relaxedpeak) */
      if( relaxedpeak < inferpeak )
      {
         inferpeak = MAX(maxlst, relaxedpeak);
      }
      else if( relaxedpeak > inferpeak )
      {
         inferpeak = MIN(minect-1, relaxedpeak);
      }
      assert(inferpeak >= hmin);
      assert(inferpeak < hmax);
      assert(inferpeak >= maxlst);
      assert(inferpeak < minect);

      /* post all necessary bound changes */
      for( c = 0; c < ncands; ++c )
      {
         var = vars[cands[c]];
         assert(var != NULL);

         if( usebdwidening )
         {
            duration = durations[cands[c]];

            SCIP_CALL( SCIPaddConflictRelaxedLb(scip, var, bdchgidx, (SCIP_Real)(inferpeak - duration + 1)) );
            SCIP_CALL( SCIPaddConflictRelaxedUb(scip, var, bdchgidx, (SCIP_Real)inferpeak) );
         }
         else
         {
            SCIP_CALL( SCIPaddConflictLb(scip, var, bdchgidx) );
            SCIP_CALL( SCIPaddConflictUb(scip, var, bdchgidx) );
         }

         if( explanation != NULL )
            explanation[cands[c]] = TRUE;
      }

      SCIPfreeBufferArray(scip, &canddemands);
      SCIPfreeBufferArray(scip, &cands);
   }

   SCIPfreeBufferArray(scip, &reported);

   if( provedpeak != NULL )
      *provedpeak = inferpeak;

   return SCIP_OKAY;
}

#if 0
/** repropagation of edge finding algorithm simplified version from Petr Vilim only a small subset is reported such that
 *  energy in total and for bound change is enough
 */
static
SCIP_RETCODE resolvePropagationEdgeFinding(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             infervar,           /**< variable whose bound change is to be explained */
   INFERINFO             inferinfo,          /**< inference info containing position of correct bdchgids */
   SCIP_BOUNDTYPE        boundtype,          /**< the type of the changed bound (lower or upper bound) */
   SCIP_BDCHGIDX*        bdchgidx,           /**< the index of the bound change, representing the point of time where the change took place */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            explanation         /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   )
{
   int est;
   int lct;
   int j;

   /* ???????????????????? do bound widening */

   assert(scip != NULL);
   assert(nvars > 0);
   assert(inferInfoGetProprule(inferinfo) == PROPRULE_2_EDGEFINDING);

   SCIPdebugMessage("repropagate edge-finding with short reasons for variable <%s>\n", SCIPvarGetName(infervar));

   if( boundtype == SCIP_BOUNDTYPE_LOWER )
   {
      SCIP_CALL( SCIPaddConflictLb(scip, infervar, bdchgidx) );
   }
   else
   {
      SCIP_CALL( SCIPaddConflictUb(scip, infervar, bdchgidx) );
   }

   est = inferInfoGetData1(inferinfo);
   lct = inferInfoGetData2(inferinfo);
   assert(est < lct);

   /* collect the energies of all variables in [est_omega, lct_omega] */
   for( j = 0; j < nvars; ++j )
   {
      SCIP_VAR* var;
      SCIP_Bool left;
      SCIP_Bool right;
      int duration;
      int lb;
      int ub;

      var = vars[j];
      assert(var != NULL);

      if( var == infervar )
      {
         if( explanation != NULL )
            explanation[j] = TRUE;

         continue;
      }

      lb = SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE));
      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE));

      duration = durations[j];
      assert(duration > 0);

      /* in case the earliest start time is equal to hmin we have to also consider the jobs which run in that region
       * since we use adjusted jobs during the propagation
       */
      left = (est == hmin && lb + duration > hmin) || lb >= est;

      /* in case the latest completion time is equal to hmax we have to also consider the jobs which run in that region
       * since we use adjusted jobs during the propagation
       */
      right = (lct == hmax && ub < hmax) || ub + duration <= lct;

      /* store all jobs running in [est_omega; lct_omega] */
      if( left && right  )
      {
         /* check if bound widening should be used */
         if( usebdwidening )
         {
            SCIP_CALL( SCIPaddConflictRelaxedLb(scip, var, bdchgidx, (SCIP_Real)(lct - duration)) );
            SCIP_CALL( SCIPaddConflictRelaxedUb(scip, var, bdchgidx, (SCIP_Real)(est)) );
         }
         else
         {
            SCIP_CALL( SCIPaddConflictLb(scip, var, bdchgidx) );
            SCIP_CALL( SCIPaddConflictUb(scip, var, bdchgidx) );
         }

         if( explanation != NULL )
            explanation[j] = TRUE;
      }
   }

   return SCIP_OKAY;
}
#endif

/** compute the minimum overlaps w.r.t. the duration of the job and the time window [begin,end) */
static
int computeOverlap(
   int                   begin,              /**< begin of the times interval */
   int                   end,                /**< end of time interval */
   int                   est,                /**< earliest start time */
   int                   lst,                /**< latest start time */
   int                   duration            /**< duration of the job */
   )
{
   int left;
   int right;
   int ect;
   int lct;

   ect = est + duration;
   lct = lst + duration;

   /* check if job runs completely within [begin,end) */
   if( lct <= end && est >= begin )
      return duration;

   assert(lst <= end && ect >= begin);

   left = ect - begin;
   assert(left > 0);

   right = end - lst;
   assert(right > 0);

   return MIN3(left, right, end - begin);
}

/** an overload was detected due to the time-time edge-finding propagate; initialized conflict analysis, add an initial
 *  reason
 *
 *  @note the conflict analysis is not performend, only the initialized SCIP_Bool pointer is set to TRUE
 */
static
SCIP_RETCODE analyzeEnergyRequirement(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< capacity of the cumulative condition */
   int                   begin,              /**< begin of the time window */
   int                   end,                /**< end of the time window */
   SCIP_VAR*             infervar,           /**< variable which was propagate, or NULL */
   SCIP_BOUNDTYPE        boundtype,          /**< the type of the changed bound (lower or upper bound) */
   SCIP_BDCHGIDX*        bdchgidx,           /**< the index of the bound change, representing the point of time where the change took place */
   SCIP_Real             relaxedbd,          /**< the relaxed bound which is sufficient to be explained */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            explanation         /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   )
{
   int* locenergies;
   int* overlaps;
   int* idxs;

   int requiredenergy;
   int v;

   SCIP_CALL( SCIPallocBufferArray(scip, &locenergies, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &overlaps, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &idxs, nvars) );

   /* energy which needs be explained */
   requiredenergy = (end - begin) * capacity;

   SCIPdebugMessage("analysis energy load in [%d,%d) (capacity %d, energy %d)\n", begin, end, capacity, requiredenergy);

   /* collect global contribution and adjusted the required energy by the amount of energy the inference variable
    * takes
    */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      int glbenergy;
      int duration;
      int demand;
      int est;
      int lst;

      var = vars[v];
      assert(var != NULL);

      locenergies[v] = 0;
      overlaps[v] = 0;
      idxs[v] = v;

      demand = demands[v];
      assert(demand > 0);

      duration = durations[v];
      assert(duration > 0);

      /* check if the variable equals the inference variable (the one which was propagated) */
      if( infervar == var )
      {
         int overlap;
         int right;
         int left;

         assert(relaxedbd != SCIP_UNKNOWN); /*lint !e777*/

         SCIPdebugMessage("inference variable <%s>[%g,%g] %s %g (duration %d, demand %d)\n",
            SCIPvarGetName(var), SCIPvarGetLbAtIndex(var, bdchgidx, FALSE), SCIPvarGetUbAtIndex(var, bdchgidx, FALSE),
            boundtype == SCIP_BOUNDTYPE_LOWER ? ">=" : "<=", relaxedbd, duration, demand);

         /* compute the amount of energy which needs to be available for enforcing the propagation and report the bound
          * which is necessary from the inference variable
          */
         if( boundtype == SCIP_BOUNDTYPE_UPPER )
         {
            int lct;

            /* get the latest start time of the infer start time variable before the propagation took place */
            lst = SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE));

            /* the latest start time of the inference start time variable before the propagation needs to be smaller as
             * the end of the time interval; meaning the job needs be overlap with the time interval in case the job is
             * scheduled w.r.t. its latest start time
             */
            assert(lst < end);

            /* compute the overlap of the job in case it would be scheduled w.r.t. its latest start time and the time
             * interval (before the propagation)
             */
            right = MIN3(end - lst, end - begin, duration);

            /* the job needs to overlap with the interval; otherwise the propagation w.r.t. this time window is not valid */
            assert(right > 0);

            lct = SCIPconvertRealToInt(scip, relaxedbd) + duration;
            assert(begin <= lct);
            assert(bdchgidx == NULL || SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, TRUE)) < begin);

            /* compute the overlap of the job after the propagation but considering the relaxed bound */
            left = MIN(lct - begin + 1, end - begin);
            assert(left > 0);

            /* compute the minimum overlap; */
            overlap = MIN(left, right);
            assert(overlap > 0);
            assert(overlap <= end - begin);
            assert(overlap <= duration);

            if( usebdwidening )
            {
               assert(SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE)) <= (end - overlap));
               SCIP_CALL( SCIPaddConflictRelaxedUb(scip, var, bdchgidx, (SCIP_Real)(end - overlap)) );
            }
            else
            {
               SCIP_CALL( SCIPaddConflictUb(scip, var, bdchgidx) );
            }
         }
         else
         {
            int ect;

            assert(boundtype == SCIP_BOUNDTYPE_LOWER);

            /* get the earliest completion time of the infer start time variable before the propagation took place */
            ect = SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)) + duration;

            /* the earliest start time of the inference start time variable before the propagation needs to be larger as
             * than the beginning of the time interval; meaning the job needs be overlap with the time interval in case
             * the job is scheduled w.r.t. its earliest start time
             */
            assert(ect > begin);

            /* compute the overlap of the job in case it would be scheduled w.r.t. its earliest start time and the time
             * interval (before the propagation)
             */
            left = MIN3(ect - begin, end - begin, duration);

            /* the job needs to overlap with the interval; otherwise the propagation w.r.t. this time window is not valid */
            assert(left > 0);

            est = SCIPconvertRealToInt(scip, relaxedbd);
            assert(end >= est);
            assert(bdchgidx == NULL || end - SCIPvarGetLbAtIndex(var, bdchgidx, TRUE) < duration);

            /* compute the overlap of the job after the propagation but considering the relaxed bound */
            right = MIN(end - est + 1, end - begin);
            assert(right > 0);

            /* compute the minimum overlap */
            overlap = MIN(left, right);
            assert(overlap > 0);
            assert(overlap <= end - begin);
            assert(overlap <= duration);

            if( usebdwidening )
            {
               assert(SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE)) >= (begin + overlap - duration));
               SCIP_CALL( SCIPaddConflictRelaxedLb(scip, var, bdchgidx, (SCIP_Real)(begin + overlap - duration)) );
            }
            else
            {
               SCIP_CALL( SCIPaddConflictLb(scip, var, bdchgidx) );
            }
         }

         /* subtract the amount of energy which is available due to the overlap of the inference start time */
         requiredenergy -=  overlap * demand;

         if( explanation != NULL )
            explanation[v] = TRUE;

         continue;
      }

      /* global time points */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));

      glbenergy = 0;

      /* check if the has any overlap w.r.t. global bound; meaning some parts of the job will run for sure within the
       * time window
       */
      if( est + duration > begin && lst < end )
      {
         /* evaluated global contribution */
         glbenergy = computeOverlap(begin, end, est, lst, duration) * demand;

         /* remove the globally available energy form the required energy */
         requiredenergy -= glbenergy;

         if( explanation != NULL )
            explanation[v] = TRUE;
      }

      /* local time points */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(var, bdchgidx, FALSE));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(var, bdchgidx, FALSE));

      /* check if the job has any overlap w.r.t. local bound; meaning some parts of the job will run for sure within the
       * time window
       */
      if( est + duration > begin && lst < end )
      {
         overlaps[v] = computeOverlap(begin, end, est, lst, duration);

         /* evaluated additionally local energy contribution */
         locenergies[v] = overlaps[v] * demand - glbenergy;
         assert(locenergies[v] >= 0);
      }
   }

   /* sort the variable contributions w.r.t. additional local energy contributions */
   SCIPsortDownIntIntInt(locenergies, overlaps, idxs, nvars);

   /* add local energy contributions until an overload is implied */
   for( v = 0; v < nvars && requiredenergy >= 0; ++v )
   {
      SCIP_VAR* var;
      int duration;
      int overlap;
      int relaxlb;
      int relaxub;
      int idx;

      idx = idxs[v];
      assert(idx >= 0 && idx < nvars);

      var = vars[idx];
      assert(var != NULL);
      assert(var != infervar);

      duration = durations[idx];
      assert(duration > 0);

      overlap = overlaps[v];
      assert(overlap > 0);

      requiredenergy -= locenergies[v];

      if( requiredenergy < -1 )
      {
         int demand;

         demand = demands[idx];
         assert(demand > 0);

         overlap += (int)((requiredenergy + 1) / demand);

#ifndef NDEBUG
         requiredenergy += locenergies[v];
         requiredenergy -= overlap * demand;
         assert(requiredenergy < 0);
#endif
      }
      assert(overlap > 0);

      relaxlb = begin - duration + overlap;
      relaxub = end - overlap;

      SCIPdebugMessage("variable <%s> glb=[%g,%g] loc=[%g,%g], conf=[%g,%g], added=[%d,%d] (demand %d, duration %d)\n",
         SCIPvarGetName(var),
         SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var),
         SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var),
         SCIPgetConflictVarLb(scip, var), SCIPgetConflictVarUb(scip, var),
         relaxlb, relaxub, demands[idx], duration);

      SCIP_CALL( SCIPaddConflictRelaxedLb(scip, var, bdchgidx, (SCIP_Real)relaxlb) );
      SCIP_CALL( SCIPaddConflictRelaxedUb(scip, var, bdchgidx, (SCIP_Real)relaxub) );

      if( explanation != NULL )
         explanation[idx] = TRUE;
   }

   assert(requiredenergy < 0);

   SCIPfreeBufferArray(scip, &idxs);
   SCIPfreeBufferArray(scip, &overlaps);
   SCIPfreeBufferArray(scip, &locenergies);

   return SCIP_OKAY;
}

/** resolve propagation w.r.t. the cumulative condition */
static
SCIP_RETCODE respropCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             infervar,           /**< the conflict variable whose bound change has to be resolved */
   INFERINFO             inferinfo,          /**< the user information */
   SCIP_BOUNDTYPE        boundtype,          /**< the type of the changed bound (lower or upper bound) */
   SCIP_BDCHGIDX*        bdchgidx,           /**< the index of the bound change, representing the point of time where the change took place */
   SCIP_Real             relaxedbd,          /**< the relaxed bound which is sufficient to be explained */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_RESULT*          result              /**< pointer to store the result of the propagation conflict resolving call */
   )
{
   switch( inferInfoGetProprule(inferinfo) )
   {
   case PROPRULE_1_CORETIMES:
   {
      int inferdemand;
      int inferduration;
      int inferpos;
      int inferpeak;
      int relaxedpeak;
      int provedpeak;

      /* get the position of the inferred variable in the vars array */
      inferpos = inferInfoGetData1(inferinfo);
      if( inferpos >= nvars || vars[inferpos] != infervar )
      {
         /* find inference variable in constraint */
         for( inferpos = 0; inferpos < nvars && vars[inferpos] != infervar; ++inferpos )
         {}
      }
      assert(inferpos < nvars);
      assert(vars[inferpos] == infervar);

      inferdemand = demands[inferpos];
      inferduration = durations[inferpos];

      if( boundtype == SCIP_BOUNDTYPE_UPPER )
      {
         /* we propagated the latest start time (upper bound) step wise with a step length of at most the duration of
          * the inference variable
          */
         assert(SCIPvarGetUbAtIndex(infervar, bdchgidx, FALSE) - SCIPvarGetUbAtIndex(infervar, bdchgidx, TRUE) < inferduration + 0.5);

         SCIPdebugMessage("variable <%s>: upper bound changed from %g to %g (relaxed %g)\n",
            SCIPvarGetName(infervar), SCIPvarGetUbAtIndex(infervar, bdchgidx, FALSE),
            SCIPvarGetUbAtIndex(infervar, bdchgidx, TRUE), relaxedbd);

         /* get the inference peak that the time point which lead to the that propagtion */
         inferpeak = inferInfoGetData2(inferinfo);
         /* the bound passed back to be resolved might be tighter as the bound propagted by the core time propagator;
          * this can happen if the variable is not activ and aggregated to an activ variable with a scale != 1.0
          */
         assert(SCIPconvertRealToInt(scip, SCIPvarGetUbAtIndex(infervar, bdchgidx, TRUE)) + inferduration <= inferpeak);
         relaxedpeak = SCIPconvertRealToInt(scip, relaxedbd) + inferduration;

         /* make sure that the relaxed peak is part of the effective horizon */
         relaxedpeak = MIN(relaxedpeak, hmax-1);

         /* make sure that relaxed peak is not larger than the infer peak
          *
          * This can happen in case the variable is not an active variable!
          */
         relaxedpeak = MAX(relaxedpeak, inferpeak);
         assert(relaxedpeak >= inferpeak);
         assert(relaxedpeak >= hmin);
      }
      else
      {
         assert(boundtype == SCIP_BOUNDTYPE_LOWER);

         SCIPdebugMessage("variable <%s>: lower bound changed from %g to %g (relaxed %g)\n",
            SCIPvarGetName(infervar), SCIPvarGetLbAtIndex(infervar, bdchgidx, FALSE),
            SCIPvarGetLbAtIndex(infervar, bdchgidx, TRUE), relaxedbd);

         /* get the time interval where the job could not be scheduled */
         inferpeak = inferInfoGetData2(inferinfo);
         /* the bound passed back to be resolved might be tighter as the bound propagted by the core time propagator;
          * this can happen if the variable is not activ and aggregated to an activ variable with a scale != 1.0
          */
         assert(SCIPconvertRealToInt(scip, SCIPvarGetLbAtIndex(infervar, bdchgidx, TRUE)) - 1 >= inferpeak);
         relaxedpeak = SCIPconvertRealToInt(scip, relaxedbd) - 1;

         /* make sure that the relaxed peak is part of the effective horizon */
         relaxedpeak = MAX(relaxedpeak, hmin);

         /* make sure that relaxed peak is not larger than the infer peak
          *
          * This can happen in case the variable is not an active variable!
          */
         relaxedpeak = MIN(relaxedpeak, inferpeak);
         assert(relaxedpeak < hmax);
      }

      /* resolves the propagation of the core time algorithm */
      SCIP_CALL( resolvePropagationCoretimes(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
            infervar, inferdemand, inferpeak, relaxedpeak, bdchgidx, usebdwidening, &provedpeak, explanation) );

      if( boundtype == SCIP_BOUNDTYPE_UPPER )
      {
         if( usebdwidening )
         {
            SCIP_CALL( SCIPaddConflictRelaxedUb(scip, infervar, NULL,  (SCIP_Real)provedpeak) );
         }
         else
         {
            /* old upper bound of variable itself is part of the explanation */
            SCIP_CALL( SCIPaddConflictUb(scip, infervar, bdchgidx) );
         }
      }
      else
      {
         assert(boundtype == SCIP_BOUNDTYPE_LOWER);

         if( usebdwidening )
         {
            SCIP_CALL( SCIPaddConflictRelaxedLb(scip, infervar, bdchgidx, (SCIP_Real)(provedpeak - inferduration + 1)) );
         }
         else
         {
            /* old lower bound of variable itself is part of the explanation */
            SCIP_CALL( SCIPaddConflictLb(scip, infervar, bdchgidx) );
         }
      }

      if( explanation != NULL )
         explanation[inferpos] = TRUE;

      break;
   }
   case PROPRULE_2_EDGEFINDING:
   case PROPRULE_3_TTEF:
   {
      int begin;
      int end;

      begin = inferInfoGetData1(inferinfo);
      end = inferInfoGetData2(inferinfo);
      assert(begin < end);

      begin = MAX(begin, hmin);
      end = MIN(end, hmax);

      SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
            begin, end, infervar, boundtype, bdchgidx, relaxedbd, usebdwidening, explanation) );

      break;
   }

   default:
      SCIPerrorMessage("invalid inference information %d\n", inferInfoGetProprule(inferinfo));
      SCIPABORT();
      return SCIP_INVALIDDATA; /*lint !e527*/
   }

   (*result) = SCIP_SUCCESS;

   return SCIP_OKAY;
}

/**@} */


/**@name Enforcement methods
 *
 * @{
 */

/** apply all fixings which are given by the alternative bounds */
static
SCIP_RETCODE applyAlternativeBoundsBranching(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR**            vars,               /**< array of active variables */
   int                   nvars,              /**< number of active variables */
   int*                  alternativelbs,     /**< alternative lower bounds */
   int*                  alternativeubs,     /**< alternative lower bounds */
   int*                  downlocks,          /**< number of constraints with down lock participating by the computation */
   int*                  uplocks,            /**< number of constraints with up lock participating by the computation */
   SCIP_Bool*            branched            /**< pointer to store if a branching was applied */
   )
{
   int v;

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      SCIP_Real objval;

      var = vars[v];
      assert(var != NULL);

      objval = SCIPvarGetObj(var);

      if( SCIPvarGetNLocksDown(var) == downlocks[v] && !SCIPisNegative(scip, objval) )
      {
         int ub;

         ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

         if( alternativelbs[v] <= ub )
         {
            SCIP_CALL( SCIPbranchVarHole(scip, var, SCIPvarGetLbLocal(var), (SCIP_Real)alternativelbs[v], NULL, NULL) );
            (*branched) = TRUE;

            SCIPdebugMessage("variable <%s> branched domain hole (%g,%d)\n", SCIPvarGetName(var),
               SCIPvarGetLbLocal(var), alternativelbs[v]);

            return SCIP_OKAY;
         }
      }

      if( SCIPvarGetNLocksUp(var) == uplocks[v] && !SCIPisPositive(scip, objval) )
      {
         int lb;

         lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));

         if( alternativeubs[v] >= lb )
         {
            SCIP_CALL( SCIPbranchVarHole(scip, var, (SCIP_Real)alternativeubs[v], SCIPvarGetUbLocal(var), NULL, NULL) );
            (*branched) = TRUE;

            SCIPdebugMessage("variable <%s> branched domain hole (%d,%g)\n", SCIPvarGetName(var),
               alternativeubs[v],  SCIPvarGetUbLocal(var));

            return SCIP_OKAY;
         }
      }
   }

   return SCIP_OKAY;
}

/** remove the capacity requirments for all job which start at the curtime */
static
void subtractStartingJobDemands(
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   int                   curtime,            /**< current point in time */
   int*                  starttimes,         /**< array of start times */
   int*                  startindices,       /**< permutation with respect to the start times */
   int*                  freecapacity,       /**< pointer to store the resulting free capacity */
   int*                  idx,                /**< pointer to index in start time array */
   int                   nvars               /**< number of vars in array of starttimes and startindices */
   )
{

#if defined SCIP_DEBUG && !defined NDEBUG
   int oldidx;
   oldidx = *idx;
#endif

   assert(idx != NULL);
   assert(starttimes != NULL);
   assert(starttimes != NULL);
   assert(freecapacity != NULL);
   assert(starttimes[*idx] == curtime);
   assert(consdata->demands != NULL);
   assert(freecapacity != idx);

   /* subtract all capacity needed up to this point */
   (*freecapacity) -= consdata->demands[startindices[*idx]];

   while( (*idx)+1 < nvars && starttimes[(*idx)+1] == curtime )
   {
      ++(*idx);
      (*freecapacity) -= consdata->demands[startindices[(*idx)]];
      assert(freecapacity != idx);
   }
#ifdef SCIP_DEBUG
   assert(oldidx <= *idx);
#endif
}

/** add the capacity requirments for all job which end at the curtime */
static
void addEndingJobDemands(
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   int                   curtime,            /**< current point in time */
   int*                  endtimes,           /**< array of end times */
   int*                  endindices,         /**< permutation with rspect to the end times */
   int*                  freecapacity,       /**< pointer to store the resulting free capacity */
   int*                  idx,                /**< pointer to index in end time array */
   int                   nvars               /**< number of vars in array of starttimes and startindices */
   )
{
#if defined SCIP_DEBUG && !defined NDEBUG
   int oldidx;
   oldidx = *idx;
#endif

   /* free all capacity usages of jobs the are no longer running */
   while( endtimes[*idx] <= curtime && *idx < nvars)
   {
      (*freecapacity) += consdata->demands[endindices[*idx]];
      ++(*idx);
   }

#ifdef SCIP_DEBUG
   assert(oldidx <= *idx);
#endif
}

/** computes a point in time when the capacity is exceeded returns hmax if this does not happen */
static
SCIP_RETCODE computePeak(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint handler data */
   SCIP_SOL*             sol,                /**< primal solution, or NULL for current LP/pseudo solution */
   int*                  timepoint           /**< pointer to store the time point of the peak */
   )
{
   int* starttimes;         /* stores when each job is starting */
   int* endtimes;           /* stores when each job ends */
   int* startindices;       /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;         /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   int nvars;               /* number of activities for this constraint */
   int freecapacity;        /* remaining capacity */
   int curtime;             /* point in time which we are just checking */
   int endindex;            /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int hmin;
   int hmax;

   int j;

   assert(consdata != NULL);

   nvars = consdata->nvars;
   assert(nvars > 0);

   *timepoint = consdata->hmax;

   assert(consdata->vars != NULL);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   /* create event point arrays */
   createSortedEventpointsSol(scip, sol, consdata->nvars, consdata->vars, consdata->durations,
      starttimes, endtimes, startindices, endindices);

   endindex = 0;
   freecapacity = consdata->capacity;
   hmin = consdata->hmin;
   hmax = consdata->hmax;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      curtime = starttimes[j];
      SCIPdebugMessage("look at %d-th job with start %d\n", j, curtime);

      if( curtime >= hmax )
         break;

      /* remove the capacity requirments for all job which start at the curtime */
      subtractStartingJobDemands(consdata, curtime, starttimes, startindices, &freecapacity, &j, nvars);

      /* add the capacity requirments for all job which end at the curtime */
      addEndingJobDemands(consdata, curtime, endtimes, endindices, &freecapacity, &endindex, nvars);

      assert(freecapacity <= consdata->capacity);
      assert(endindex <= nvars);

      /* endindex - points to the next job which will finish */
      /* j - points to the last job that has been released */

      /* if free capacity is smaller than zero, then add branching candidates */
      if( freecapacity < 0 && curtime >= hmin )
      {
         *timepoint = curtime;
         break;
      }
   } /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   return SCIP_OKAY;
}

/** checks all cumulative constraints for infeasibility and add branching candidates to storage */
static
SCIP_RETCODE collectBranchingCands(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           conss,              /**< constraints to be processed */
   int                   nconss,             /**< number of constraints */
   SCIP_SOL*             sol,                /**< primal solution, or NULL for current LP/pseudo solution */
   int*                  nbranchcands        /**< pointer to store the number of branching variables */
   )
{
   SCIP_HASHTABLE* collectedvars;
   int c;

   assert(scip != NULL);
   assert(conss != NULL);

   /* create a hash table */
   SCIP_CALL( SCIPhashtableCreate(&collectedvars, SCIPblkmem(scip), SCIPcalcHashtableSize(SCIPgetNVars(scip)),
         SCIPvarGetHashkey, SCIPvarIsHashkeyEq, SCIPvarGetHashkeyVal, NULL) );

   assert(scip != NULL);
   assert(conss != NULL);

   for( c = 0; c < nconss; ++c )
   {
      SCIP_CONS* cons;
      SCIP_CONSDATA* consdata;

      int curtime;
      int j;

      cons = conss[c];
      assert(cons != NULL);

      if( !SCIPconsIsActive(cons) )
         continue;

      consdata = SCIPconsGetData(cons);
      assert(consdata != NULL);

      /* get point in time when capacity is exceeded */
      SCIP_CALL( computePeak(scip, consdata, sol, &curtime) );

      if( curtime < consdata->hmin || curtime >= consdata->hmax )
         continue;

      /* report all variables that are running at that point in time */
      for( j = 0; j < consdata->nvars; ++j )
      {
         SCIP_VAR* var;
         int lb;
         int ub;

         var = consdata->vars[j];
         assert(var != NULL);

         /* check if the variable was already added */
         if( SCIPhashtableExists(collectedvars, (void*)var) )
            continue;

         lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
         ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

         if( lb <= curtime && ub + consdata->durations[j] > curtime && lb < ub  )
         {
            SCIP_Real solval;
            SCIP_Real score;

            solval = SCIPgetSolVal(scip, sol, var);
            score = MIN(solval - lb, ub - solval) / ((SCIP_Real)ub-lb);

            SCIPdebugMessage("add var <%s> to branch cand storage\n", SCIPvarGetName(var));
            SCIP_CALL( SCIPaddExternBranchCand(scip, var, score, lb + (ub - lb) / 2.0 + 0.2) );
            (*nbranchcands)++;

            SCIP_CALL( SCIPhashtableInsert(collectedvars, var) );
         }
      }
   }

   SCIPhashtableFree(&collectedvars);

   SCIPdebugMessage("found %d branching candidates\n", *nbranchcands);

   return SCIP_OKAY;
}

/** enforcement pseudo or LP solution */
static
SCIP_RETCODE enforceSolution(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           conss,              /**< constraints to be processed */
   int                   nconss,             /**< number of constraints */
   SCIP_Bool             branch,             /**< should branching candidates be collected */
   SCIP_RESULT*          result              /**< pointer to store the result */
   )
{
   if( branch )
   {
      int nbranchcands;

      nbranchcands = 0;
      SCIP_CALL( collectBranchingCands(scip, conss, nconss, NULL, &nbranchcands) );

      if( nbranchcands > 0 )
         (*result) = SCIP_INFEASIBLE;
   }
   else
   {
      SCIP_Bool violated;
      int c;

      violated = FALSE;

      /* first check if a constraints is violated */
      for( c = 0; c < nconss && !violated; ++c )
      {
         SCIP_CONS* cons;

         cons = conss[c];
         assert(cons != NULL);

         SCIP_CALL( checkCons(scip, cons, NULL, &violated, FALSE) );
      }

      if( violated )
         (*result) = SCIP_INFEASIBLE;
   }

   return SCIP_OKAY;
}

/**@} */

/**@name Propagation
 *
 * @{
 */

/** check if cumulative constraint is independently of all other constraints */
static
SCIP_Bool isConsIndependently(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< cumulative constraint */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR** vars;
   SCIP_Bool* downlocks;
   SCIP_Bool* uplocks;
   int nvars;
   int v;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;
   vars = consdata->vars;
   downlocks = consdata->downlocks;
   uplocks = consdata->uplocks;

   /* check if the cumulative constraint has the only locks on the involved variables */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;

      var = vars[v];
      assert(var != NULL);

      if( SCIPvarGetNLocksDown(var) > (int)downlocks[v] || SCIPvarGetNLocksUp(var) > (int)uplocks[v] )
         return FALSE;
   }

   return TRUE;
}

/** in case the cumulative constraint is independent of every else, solve the cumulative problem and apply the fixings
 *  (dual reductions)
 */
static
SCIP_RETCODE solveIndependentCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   SCIP_Longint          maxnodes,           /**< number of branch-and-bound nodes to solve an independent cumulative constraint  (-1: no limit) */
   int*                  nchgbds,            /**< pointer to store the number changed variable bounds */
   int*                  nfixedvars,         /**< pointer to count number of fixings */
   int*                  ndelconss,          /**< pointer to count number of deleted constraints  */
   SCIP_Bool*            cutoff,             /**< pointer to store if the constraint is infeasible */
   SCIP_Bool*            unbounded           /**< pointer to store if the constraint is unbounded */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR** vars;
   SCIP_Real* objvals;
   SCIP_Real* lbs;
   SCIP_Real* ubs;
   SCIP_Real timelimit;
   SCIP_Real memorylimit;
   SCIP_Bool solved;
   SCIP_Bool error;

   int ncheckconss;
   int nvars;
   int v;

   assert(scip != NULL);
   assert(!SCIPconsIsModifiable(cons));
   assert(SCIPgetNConss(scip) > 0);

   /* if SCIP is in probing mode or repropagation we cannot perform this dual reductions since this dual reduction
    * would/could end in an implication which can lead to cutoff of the/all optimal solution
    */
   if( SCIPinProbing(scip) || SCIPinRepropagation(scip) )
      return SCIP_OKAY;

   /* constraints for which the check flag is set to FALSE, did not contribute to the lock numbers; therefore, we cannot
    * use the locks to decide for a dual reduction using this constraint;
    */
   if( !SCIPconsIsChecked(cons) )
      return SCIP_OKAY;

   ncheckconss = SCIPgetNCheckConss(scip);

   /* if the cumulative constraint is the only constraint of the original problem or the only check constraint in the
    * presolved problem do nothing execpt to change the parameter settings
    */
   if( ncheckconss == 1 )
   {
      /* shrink the minimal maximum value for the conflict length */
      SCIP_CALL( SCIPsetIntParam(scip, "conflict/minmaxvars", 10) );

      /* use only first unique implication point */
      SCIP_CALL( SCIPsetIntParam(scip, "conflict/fuiplevels", 1) );

      /* do not use reconversion conflicts */
      SCIP_CALL( SCIPsetIntParam(scip, "conflict/reconvlevels", 0) );

      /* after 250 conflict we force a restart since then the variable statistics are reasonable initialized */
      SCIP_CALL( SCIPsetIntParam(scip, "conflict/restartnum", 250) );

      /* increase the number of conflicts which induce a restart */
      SCIP_CALL( SCIPsetRealParam(scip, "conflict/restartfac", 2.0) );

      /* weight the variable which made into a conflict */
      SCIP_CALL( SCIPsetRealParam(scip, "conflict/conflictweight", 1.0) );

      /* do not check pseudo solution (for performance reasons) */
      SCIP_CALL( SCIPsetBoolParam(scip, "constraints/disableenfops", TRUE) );

      /* use value based history to detect a reasonable branching point */
      SCIP_CALL( SCIPsetBoolParam(scip, "history/valuebased", TRUE) );

      /* turn of LP relaxation */
      SCIP_CALL( SCIPsetIntParam(scip, "lp/solvefreq", -1) );

      /* prefer the down branch in case the value based history does not suggest something */
      SCIP_CALL( SCIPsetCharParam(scip, "nodeselection/childsel", 'd') );

      /* accept any bound change */
      SCIP_CALL( SCIPsetRealParam(scip, "numerics/boundstreps", 1e-6) );

      /* allow for at most 10 restart, after that the value based history should be reliable */
      SCIP_CALL( SCIPsetIntParam(scip, "presolving/maxrestarts", 10) );

      /* set priority for depth first search to highest possible value */
      SCIP_CALL( SCIPsetIntParam(scip, "nodeselection/dfs/stdpriority", INT_MAX/4) );

      return SCIP_OKAY;
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   /* check if already tried to solve that constraint as independent sub problem; we do not want to try it again if we
    * fail on the first place
    */
   if( consdata->triedsolving )
      return SCIP_OKAY;

   /* check if constraint is independently */
   if( !isConsIndependently(scip, cons) )
      return SCIP_OKAY;

   /* mark the constraint to be tried of solving it as independent sub problem; in case that is successful the
    * constraint is deleted; otherwise, we want to ensure that we do not try that again
    */
   consdata->triedsolving = TRUE;

   SCIPdebugMessage("the cumulative constraint <%s> is independent from rest of the problem (%d variables, %d constraints)\n",
      SCIPconsGetName(cons), SCIPgetNVars(scip), SCIPgetNConss(scip));
   SCIPdebugPrintCons(scip, cons, NULL);

   nvars = consdata->nvars;
   vars = consdata->vars;

   SCIP_CALL( SCIPallocBufferArray(scip, &lbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &ubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &objvals, nvars) );

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;

      /* if a variables array is given, use the variable bounds otherwise the default values stored in the ests and lsts
       * array
       */
      var = vars[v];
      assert(var != NULL);

      lbs[v] = SCIPvarGetLbLocal(var);
      ubs[v] = SCIPvarGetUbLocal(var);

      objvals[v] = SCIPvarGetObj(var);
   }

   /* check whether there is enough time and memory left */
   SCIP_CALL( SCIPgetRealParam(scip, "limits/time", &timelimit) );
   if( !SCIPisInfinity(scip, timelimit) )
      timelimit -= SCIPgetSolvingTime(scip);
   SCIP_CALL( SCIPgetRealParam(scip, "limits/memory", &memorylimit) );

   /* substract the memory already used by the main SCIP and the estimated memory usage of external software */
   if( !SCIPisInfinity(scip, memorylimit) )
   {
      memorylimit -= SCIPgetMemUsed(scip)/1048576.0;
      memorylimit -= SCIPgetMemExternEstim(scip)/1048576.0;
   }

   /* solve the cumulative condition separately */
   SCIP_CALL( SCIPsolveCumulative(scip, nvars, lbs, ubs, objvals, consdata->durations, consdata->demands, consdata->capacity,
         consdata->hmin, consdata->hmax, timelimit, memorylimit, maxnodes, &solved, cutoff, unbounded, &error) );

   if( !(*cutoff) && !(*unbounded) && !error )
   {
      SCIP_Bool infeasible;
      SCIP_Bool tightened;
      SCIP_Bool allfixed;

      allfixed = TRUE;

      for( v = 0; v < nvars; ++v )
      {
         /* check if variable is fixed */
         if( lbs[v] + 0.5 > ubs[v] )
         {
            SCIP_CALL( SCIPfixVar(scip, vars[v], lbs[v], &infeasible, &tightened) );
            assert(!infeasible);

            if( tightened )
            {
               (*nfixedvars)++;
               consdata->triedsolving = FALSE;
            }
         }
         else
         {
            SCIP_CALL( SCIPtightenVarLb(scip, vars[v], lbs[v], TRUE, &infeasible, &tightened) );
            assert(!infeasible);

            if( tightened )
            {
               (*nchgbds)++;
               consdata->triedsolving = FALSE;
            }

            SCIP_CALL( SCIPtightenVarUb(scip, vars[v], ubs[v], TRUE, &infeasible, &tightened) );
            assert(!infeasible);

            if( tightened )
            {
               (*nchgbds)++;
               consdata->triedsolving = FALSE;
            }

            allfixed = FALSE;
         }
      }

      /* if all variables are fixed, remove the cumulative constraint since it is redundant */
      if( allfixed )
      {
         SCIP_CALL( SCIPdelConsLocal(scip, cons) );
         (*ndelconss)++;
      }
   }

   SCIPfreeBufferArray(scip, &objvals);
   SCIPfreeBufferArray(scip, &ubs);
   SCIPfreeBufferArray(scip, &lbs);

   return SCIP_OKAY;
}

/** start conflict analysis to analysis the core insertion which is infeasible */
static
SCIP_RETCODE analyseInfeasibelCoreInsertion(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             infervar,           /**< start time variable which lead to the infeasibilty */
   int                   inferduration,      /**< duration of the start time variable */
   int                   inferdemand,        /**< demand of the start time variable */
   int                   inferpeak,          /**< profile preak which causes the infeasibilty */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            initialized,        /**< pointer to store if the conflict analysis was initialized */
   SCIP_Bool*            explanation         /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   )
{
   SCIPdebugMessage("detected infeasibility due to adding a core to the core resource profile\n");
   SCIPdebugMessage("variable <%s>[%g,%g] (demand %d, duration %d)\n", SCIPvarGetName(infervar),
      SCIPvarGetLbLocal(infervar), SCIPvarGetUbLocal(infervar), inferdemand, inferduration);

   /* initialize conflict analysis if conflict analysis is applicable */
   if( SCIPisConflictAnalysisApplicable(scip) )
   {
      SCIP_CALL( SCIPinitConflictAnalysis(scip) );

      SCIP_CALL( resolvePropagationCoretimes(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
            infervar, inferdemand, inferpeak, inferpeak, NULL, usebdwidening, NULL, explanation) );

      SCIPdebugMessage("add lower and upper bounds of variable <%s>\n", SCIPvarGetName(infervar));

      /* add both bound of the inference variable since these biuld the core which we could not inserted */
      if( usebdwidening )
      {
         SCIP_CALL( SCIPaddConflictRelaxedLb(scip, infervar, NULL, (SCIP_Real)(inferpeak - inferduration + 1)) );
         SCIP_CALL( SCIPaddConflictRelaxedUb(scip, infervar, NULL,  (SCIP_Real)inferpeak) );
      }
      else
      {
         SCIP_CALL( SCIPaddConflictLb(scip, infervar, NULL) );
         SCIP_CALL( SCIPaddConflictUb(scip, infervar, NULL) );
      }

      *initialized = TRUE;
   }

   return SCIP_OKAY;
}

/** We are using the core resource profile which contains all core except the one of the start time variable which we
 *  want to propagate, to incease the earliest start time. This we are doing in steps of length at most the duration of
 *  the job. The reason for that is, that this makes it later easier to resolve this propagation during the conflict
 *  analysis
 */
static
SCIP_RETCODE coretimesUpdateLb(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated */
   SCIP_PROFILE*         profile,            /**< resource profile */
   int                   idx,                /**< position of the variable to propagate */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            infeasible          /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_VAR* var;
   int ntimepoints;
   int duration;
   int demand;
   int peak;
   int newlb;
   int est;
   int lst;
   int pos;

   var = vars[idx];
   assert(var != NULL);

   duration = durations[idx];
   assert(duration > 0);

   demand = demands[idx];
   assert(demand > 0);

   est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
   lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));
   ntimepoints = SCIPprofileGetNTimepoints(profile);

   /* first we find left position of earliest start time (lower bound) in resource profile; this position gives us the
    * load which we have at the earliest start time (lower bound)
    */
   (void) SCIPprofileFindLeft(profile, est, &pos);

   SCIPdebugMessage("propagate earliest start time (lower bound) (pos %d)\n", pos);

   /* we now trying to move the earliest start time in steps of at most "duration" length */
   do
   {
      INFERINFO inferinfo;
      SCIP_Bool tightened;
      int ect;

#ifndef NDEBUG
      {
         /* in debug mode we check that we adjust the search position correctly */
         int tmppos;

         (void)SCIPprofileFindLeft(profile, est, &tmppos);
         assert(pos == tmppos);
      }
#endif
      ect = est + duration;
      peak = -1;

      /* we search for a peak within the core profile which conflicts with the demand of the start time variable; we
       * want a peak which is closest to the earliest completion time
       */
      do
      {
         /* check if the profile load conflicts with the demand of the start time variable */
         if( SCIPprofileGetLoad(profile, pos) + demand > capacity )
            peak = pos;

         pos++;
      }
      while( pos < ntimepoints && SCIPprofileGetTime(profile, pos) < ect );

      /* if we found no peak that means current the job could be scheduled at its earliest start time without
       * conflicting to the core resource profile
       */
      if( peak == -1 )
         break;

      /* the peak position gives us a time point where the start time variable is in conflict with the resource
       * profile. That means we have to move it to the next time point in the resource profile but at most to the
       * earliest completion time (the remaining move will done in the next loop)
       */
      newlb = SCIPprofileGetTime(profile, peak+1);
      newlb = MIN(newlb, ect);

      /* if the earliest start time is greater than the lst we detected an infeasibilty */
      if( newlb > lst )
      {
         SCIPdebugMessage("variable <%s>: cannot be scheduled\n", SCIPvarGetName(var));

         /* use conflict analysis to analysis the core insertion which was infeasible */
         SCIP_CALL( analyseInfeasibelCoreInsertion(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
               var, duration, demand, newlb-1, usebdwidening, initialized, explanation) );

         if( explanation != NULL )
            explanation[idx] = TRUE;

         *infeasible = TRUE;

         break;
      }

      /* construct the inference information which we are using with the conflict analysis to resolve that particular
       * bound change
       */
      inferinfo = getInferInfo(PROPRULE_1_CORETIMES, idx, newlb-1);

      /* perform the bound lower bound change */
      SCIP_CALL( SCIPinferVarLbCons(scip, var, (SCIP_Real)newlb, cons, inferInfoToInt(inferinfo), TRUE, infeasible, &tightened) );
      assert(tightened);
      assert(!(*infeasible));

      SCIPdebugMessage("variable <%s> new lower bound <%d> -> <%d>\n", SCIPvarGetName(var), est, newlb);
      (*nchgbds)++;

      /* for the statistic we count the number of times a lower bound was tightened due the the time-table algorithm */
      SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nlbtimetable++ );

      /* adjust the earliest start time
       *
       * @note We are taking the lower of the start time variable on purpose instead of newlb. This is due the fact that
       *       the proposed lower bound might be even strength by be the core which can be the case if aggregations are
       *       involved.
       */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      assert(est >= newlb);

      /* adjust the search position for the resource profile for the next step */
      if( est == SCIPprofileGetTime(profile, peak+1) )
         pos = peak + 1;
      else
         pos = peak;
   }
   while( est < lst );

   return SCIP_OKAY;
}

/** We are using the core resource profile which contains all core except the one of the start time variable which we
 *  want to propagate, to decrease the latest start time. This we are doing in steps of length at most the duration of
 *  the job. The reason for that is, that this makes it later easier to resolve this propagation during the conflict
 *  analysis
 */
static
SCIP_RETCODE coretimesUpdateUb(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< start time variable to propagate */
   int                   duration,           /**< duration of the job */
   int                   demand,             /**< demand of the job */
   int                   capacity,           /**< cumulative capacity */
   SCIP_CONS*            cons,               /**< constraint which is propagated */
   SCIP_PROFILE*         profile,            /**< resource profile */
   int                   idx,                /**< position of the variable to propagate */
   int*                  nchgbds             /**< pointer to store the number of bound changes */
   )
{
   int ntimepoints;
   int newub;
   int peak;
   int pos;
   int est;
   int lst;
   int lct;

   assert(var != NULL);
   assert(duration > 0);
   assert(demand > 0);

   est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
   lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

   /* in case the start time variable is fixed do nothing */
   if( est == lst )
      return SCIP_OKAY;

   ntimepoints = SCIPprofileGetNTimepoints(profile);

   lct = lst + duration;

   /* first we find left position of latest completion time minus 1 (upper bound + duration) in resource profile; That
    * is the last time point where the job would run if schedule it at its latest start time (upper bound). This
    * position gives us the load which we have at the latest completion time minus one
    */
   (void) SCIPprofileFindLeft(profile, lct - 1, &pos);

   SCIPdebugMessage("propagate upper bound (pos %d)\n", pos);
   SCIPdebug( SCIPprofilePrint(profile, SCIPgetMessagehdlr(scip), NULL) );

   if( pos == ntimepoints-1 && SCIPprofileGetTime(profile, pos) == lst )
      return SCIP_OKAY;

   /* we now trying to move the latest start time in steps of at most "duration" length */
   do
   {
      INFERINFO inferinfo;
      SCIP_Bool tightened;
      SCIP_Bool infeasible;

      peak = -1;

#ifndef NDEBUG
      {
         /* in debug mode we check that we adjust the search position correctly */
         int tmppos;

         (void)SCIPprofileFindLeft(profile, lct - 1, &tmppos);
         assert(pos == tmppos);
      }
#endif

      /* we search for a peak within the core profile which conflicts with the demand of the start time variable; we
       * want a peak which is closest to the latest start time
       */
      do
      {
         if( SCIPprofileGetLoad(profile, pos) + demand > capacity )
            peak = pos;

         pos--;
      }
      while( pos >= 0 && SCIPprofileGetTime(profile, pos+1) > lst);

      /* if we found no peak that means the current job could be scheduled at its latest start time without conflicting
       * to the core resource profile
       */
      if( peak == -1 )
         break;

      /* the peak position gives us a time point where the start time variable is in conflict with the resource
       * profile. That means the job has be done until that point. Hence that gives us the latest completion
       * time. Note that that we want to move the bound by at most the duration length (the remaining move we are
       * doing in the next loop)
       */
      newub = SCIPprofileGetTime(profile, peak);
      newub = MAX(newub, lst) - duration;
      assert(newub >= est);

      /* construct the inference information which we are using with the conflict analysis to resolve that particular
       * bound change
       */
      inferinfo = getInferInfo(PROPRULE_1_CORETIMES, idx, newub+duration);

      /* perform the bound upper bound change */
      SCIP_CALL( SCIPinferVarUbCons(scip, var, (SCIP_Real)newub, cons, inferInfoToInt(inferinfo), TRUE, &infeasible, &tightened) );
      assert(tightened);
      assert(!infeasible);

      SCIPdebugMessage("variable <%s>: new upper bound <%d> -> <%d>\n", SCIPvarGetName(var), lst, newub);
      (*nchgbds)++;

      /* for the statistic we count the number of times a upper bound was tightened due the the time-table algorithm */
      SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nubtimetable++ );

      /* adjust the latest start and completion time
       *
       * @note We are taking the upper of the start time variable on purpose instead of newub. This is due the fact that
       *       the proposed upper bound might be even strength by be the core which can be the case if aggregations are
       *       involved.
       */
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));
      assert(lst <= newub);
      lct = lst + duration;

      /* adjust the search position for the resource profile for the next step */
      if( SCIPprofileGetTime(profile, peak) == lct )
         pos = peak - 1;
      else
         pos = peak;
   }
   while( est < lst );

   return SCIP_OKAY;
}

/** compute for the different earliest start and latest completion time the core energy of the corresponding time
 *  points
 */
static
SCIP_RETCODE computeCoreEngeryAfter(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_PROFILE*         profile,            /**< core profile */
   int                   nvars,              /**< number of start time variables (activities) */
   int*                  ests,               /**< array of sorted earliest start times */
   int*                  lcts,               /**< array of sorted latest completion times */
   int*                  coreEnergyAfterEst, /**< array to store the core energy after the earliest start time of each job */
   int*                  coreEnergyAfterLct  /**< array to store the core energy after the latest completion time of each job */
   )
{
   int ntimepoints;
   int energy;
   int t;
   int v;

   ntimepoints = SCIPprofileGetNTimepoints(profile);
   t = ntimepoints - 1;
   energy = 0;

   /* compute core energy after the earliest start time of each job */
   for( v = nvars-1; v >= 0; --v )
   {
      while( t > 0 && SCIPprofileGetTime(profile, t-1) >= ests[v] )
      {
         assert(SCIPprofileGetLoad(profile, t-1) >= 0);
         assert(SCIPprofileGetTime(profile, t) - SCIPprofileGetTime(profile, t-1)>= 0);
         energy += SCIPprofileGetLoad(profile, t-1) * (SCIPprofileGetTime(profile, t) - SCIPprofileGetTime(profile, t-1));
         t--;
      }
      assert(SCIPprofileGetTime(profile, t) >= ests[v] || t == ntimepoints-1);

      /* maybe ests[j] is in-between two timepoints */
      if( SCIPprofileGetTime(profile, t) - ests[v] > 0 )
      {
         assert(t > 0);
         coreEnergyAfterEst[v] = energy + SCIPprofileGetLoad(profile, t-1) * (SCIPprofileGetTime(profile, t) - ests[v]);
      }
      else
         coreEnergyAfterEst[v] = energy;
   }

   t = ntimepoints - 1;
   energy = 0;

   /* compute core energy after the latest completion time of each job */
   for( v = nvars-1; v >= 0; --v )
   {
      while( t > 0 && SCIPprofileGetTime(profile, t-1) >= lcts[v] )
      {
         assert(SCIPprofileGetLoad(profile, t-1) >= 0);
         assert(SCIPprofileGetTime(profile, t) - SCIPprofileGetTime(profile, t-1)>= 0);
         energy += SCIPprofileGetLoad(profile, t-1) * (SCIPprofileGetTime(profile, t) - SCIPprofileGetTime(profile, t-1));
         t--;
      }
      assert(SCIPprofileGetTime(profile, t) >= lcts[v] || t == ntimepoints-1);

      /* maybe lcts[j] is in-between two timepoints */
      if( SCIPprofileGetTime(profile, t) - lcts[v] > 0 )
      {
         assert(t > 0);
         coreEnergyAfterLct[v] = energy + SCIPprofileGetLoad(profile, t-1) * (SCIPprofileGetTime(profile, t) - lcts[v]);
      }
      else
         coreEnergyAfterLct[v] = energy;
   }

   return SCIP_OKAY;
}

/** collect earliest start times, latest completion time, and free energy contributions */
static
void collectDataTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   int*                  permests,           /**< array to store the variable positions */
   int*                  ests,               /**< array to store earliest start times */
   int*                  permlcts,           /**< array to store the variable positions */
   int*                  lcts,               /**< array to store latest completion times */
   int*                  ects,               /**< array to store earliest completion times of the flexible part of the job */
   int*                  lsts,               /**< array to store latest start times of the flexible part of the job */
   int*                  flexenergies        /**< array to store the flexible energies of each job */
   )
{
   int v;

   for( v = 0; v < nvars; ++ v)
   {
      int duration;
      int leftadjust;
      int rightadjust;
      int core;
      int est;
      int lct;
      int ect;
      int lst;

      duration = durations[v];
      assert(duration > 0);

      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[v]));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[v]));
      ect = est + duration;
      lct = lst + duration;

      ests[v] = est;
      lcts[v] = lct;
      permests[v] = v;
      permlcts[v] = v;

      /* compute core time window which lies within the effective horizon */
      core = computeCoreWithInterval(hmin, hmax, ect, lst);

      /* compute the number of time steps the job could run before the effective horizon */
      leftadjust = MAX(0, hmin - est);

      /* compute the number of time steps the job could run after the effective horizon */
      rightadjust = MAX(0, lct - hmax);

      /* compute for each job the energy which is flexible; meaning not part of the core */
      flexenergies[v] = duration - leftadjust - rightadjust - core;
      flexenergies[v] = MAX(0, flexenergies[v]);
      flexenergies[v] *= demands[v];
      assert(flexenergies[v] >= 0);

      /* the earliest completion time of the flexible energy */
      ects[v] = MIN(ect, lst);

      /* the latest start time of the flexible energy */
      lsts[v] = MAX(ect, lst);
   }
}

/** try to tighten the lower bound of the given variable */
static
SCIP_RETCODE tightenLbTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             var,                /**< variable to be considered for upper bound tightening */
   int                   duration,           /**< duration of the job */
   int                   demand,             /**< demand of the job */
   int                   est,                /**< earliest start time of the job */
   int                   ect,                /**< earliest completion time of the flexible part of the job */
   int                   lct,                /**< latest completion time of the job */
   int                   begin,              /**< begin of the time window under investigation */
   int                   end,                /**< end of the time window under investigation */
   int                   energy,             /**< available energy for the flexible part of the hob within the time window */
   int*                  bestlb,             /**< pointer to strope the best lower bound change */
   int*                  inferinfos,         /**< pointer to store the inference information which is need for the (best) lower bound change */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int newlb;

   assert(begin >= hmin);
   assert(end <= hmax);

   /* check if the time-table edge-finding should infer bounds */
   if( !conshdlrdata->ttefinfer )
      return SCIP_OKAY;

   /* if the job can be processed completely before or after the time window, nothing can be tightened */
   if( est >= end || ect <= begin )
      return SCIP_OKAY;

   /* if flexible part runs completely within the time window (assuming it is scheduled on its earliest start time), we
    * skip since the overload check will do the job
    */
   if( est >= begin && ect <= end )
      return SCIP_OKAY;

   /* check if the available energy in the time window is to small to handle the flexible part if it is schedule on its
    * earliest start time
    */
   if( energy >= demand * (MAX(begin, est) - MIN(end, ect)) )
      return SCIP_OKAY;

   /* adjust the available energy for the job; the given available energy assumes that the core of the considered job is
    * present; therefore, we need to add the core;
    *
    * @note the variable ect define the earliest completion time of the flexible part of the job; hence we need to
    *       compute the earliest completion time of the (whole) job
    */
   energy += computeCoreWithInterval(begin, end, est + duration, lct - duration) * demand;

   /* compute a latest start time (upper bound) such that the job consums at most the available energy
    *
    * @note we can round down the compute duration w.r.t. the available energy
    */
   newlb = end - energy / demand;

   /* check if we detected an infeasibility which is the case if the new lower bound is larger than the current upper
    * bound (latest start time); meaning it is not possible to schedule the job
    */
   if( newlb > lct - duration )
   {
      /* initialize conflict analysis if conflict analysis is applicable */
      if( SCIPisConflictAnalysisApplicable(scip) )
      {
         SCIP_Real relaxedbd;

         assert(SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) < newlb);

         /* it is enough to overshoot the upper bound of the variable by one */
         relaxedbd = SCIPvarGetUbLocal(var) + 1.0;

         /* initialize conflict analysis */
         SCIP_CALL( SCIPinitConflictAnalysis(scip) );

         /* added to upper bound (which was overcut be new lower bound) of the variable */
         SCIP_CALL( SCIPaddConflictUb(scip, var, NULL) );

         /* analyze the infeasible */
         SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
               begin, end, var, SCIP_BOUNDTYPE_LOWER, NULL, relaxedbd, conshdlrdata->usebdwidening, explanation) );

         (*initialized) = TRUE;
      }

      (*cutoff) = TRUE;
   }
   else if( newlb > (*bestlb) )
   {
      INFERINFO inferinfo;

      assert(newlb > begin);

      inferinfo = getInferInfo(PROPRULE_3_TTEF, begin, end);

      /* construct inference information */
      (*inferinfos) = inferInfoToInt(inferinfo);
      (*bestlb) = newlb;
   }

   return SCIP_OKAY;
}

/** try to tighten the upper bound of the given variable */
static
SCIP_RETCODE tightenUbTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             var,                /**< variable to be considered for upper bound tightening */
   int                   duration,           /**< duration of the job */
   int                   demand,             /**< demand of the job */
   int                   est,                /**< earliest start time of the job */
   int                   lst,                /**< latest start time of the flexible part of the job */
   int                   lct,                /**< latest completion time of the job */
   int                   begin,              /**< begin of the time window under investigation */
   int                   end,                /**< end of the time window under investigation */
   int                   energy,             /**< available energy for the flexible part of the hob within the time window */
   int*                  bestub,             /**< pointer to strope the best upper bound change */
   int*                  inferinfos,         /**< pointer to store the inference information which is need for the (best) upper bound change */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int newub;

   assert(begin >= hmin);
   assert(end <= hmax);
   assert(est < begin);

   /* check if the time-table edge-finding should infer bounds */
   if( !conshdlrdata->ttefinfer )
      return SCIP_OKAY;

   /* if flexible part of the job can be processed completely before or after the time window, nothing can be tightened */
   if( lst >= end || lct <= begin )
      return SCIP_OKAY;

   /* if flexible part runs completely within the time window (assuming it is scheduled on its latest start time), we
    * skip since the overload check will do the job
    */
   if( lst >= begin && lct <= end )
      return SCIP_OKAY;

   /* check if the available energy in the time window is to small to handle the flexible part  of the job */
   if( energy >= demand * (MIN(end, lct) - MAX(begin, lst)) )
      return SCIP_OKAY;

   /* adjust the available energy for the job; the given available energy assumes that the core of the considered job is
    * present; therefore, we need to add the core;
    *
    * @note the variable lst define the latest start time of the flexible part of the job; hence we need to compute the
    *       latest start of the (whole) job
    */
   energy += computeCoreWithInterval(begin, end, est + duration, lct - duration) * demand;
   assert(energy >= 0);

   /* compute a latest start time (upper bound) such that the job consums at most the available energy
    *
    * @note we can round down the compute duration w.r.t. the available energy
    */
   assert(demand > 0);
   newub = begin - duration + energy / demand;

   /* check if we detected an infeasibility which is the case if the new upper bound is smaller than the current lower
    * bound (earliest start time); meaning it is not possible to schedule the job
    */
   if( newub < est )
   {
      /* initialize conflict analysis if conflict analysis is applicable */
      if( SCIPisConflictAnalysisApplicable(scip) )
      {
         SCIP_Real relaxedbd;

         assert(SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var)) > newub);

         /* it is enough to undershoot the lower bound of the variable by one */
         relaxedbd = SCIPvarGetLbLocal(var) - 1.0;

         /* initialize conflict analysis */
         SCIP_CALL( SCIPinitConflictAnalysis(scip) );

         /* added to lower bound (which was undercut be new upper bound) of the variable */
         SCIP_CALL( SCIPaddConflictLb(scip, var, NULL) );

         /* analyze the infeasible */
         SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
               begin, end, var, SCIP_BOUNDTYPE_UPPER, NULL, relaxedbd, conshdlrdata->usebdwidening, explanation) );

         (*initialized) = TRUE;
      }

      (*cutoff) = TRUE;
   }
   else if( newub < (*bestub) )
   {
      INFERINFO inferinfo;

      assert(newub < begin);

      inferinfo = getInferInfo(PROPRULE_3_TTEF, begin, end);

      /* construct inference information */
      (*inferinfos) = inferInfoToInt(inferinfo);
      (*bestub) = newub;
   }

   return SCIP_OKAY;
}

/** propagate the upper bounds and "opportunistically" the lower bounds using the time-table edge-finding algorithm */
static
SCIP_RETCODE propagateUbTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   int*                  newlbs,             /**< array to buffer new lower bounds */
   int*                  newubs,             /**< array to buffer new upper bounds */
   int*                  lbinferinfos,       /**< array to store the inference information for the lower bound changes */
   int*                  ubinferinfos,       /**< array to store the inference information for the upper bound changes */
   int*                  lsts,               /**< array of latest start time of the flexible part in the same order as the variables */
   int*                  flexenergies,       /**< array of flexible energies in the same order as the variables */
   int*                  perm,               /**< permutation of the variables w.r.t. the non-decreasing order of the earliest start times */
   int*                  ests,               /**< array with earliest strart times sorted in non-decreasing order */
   int*                  lcts,               /**< array with latest completion times sorted in non-decreasing order */
   int*                  coreEnergyAfterEst, /**< core energy after the earliest start times */
   int*                  coreEnergyAfterLct, /**< core energy after the latest completion times */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int coreEnergyAfterEnd;
   int maxavailable;
   int minavailable;
   int totalenergy;
   int nests;
   int est;
   int lct;
   int start;
   int end;
   int v;

   est = INT_MAX;
   lct = INT_MIN;

   /* compute earliest start and latest completion time of all jobs */
   for( v = 0; v < nvars; ++v )
   {
      start = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[v]));
      end = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[v])) + durations[v];

      est = MIN(est, start);
      lct = MAX(lct, end);
   }

   /* adjust the effective time horizon */
   hmin = MAX(hmin, est);
   hmax = MIN(hmax, lct);

   end = hmax + 1;
   coreEnergyAfterEnd = -1;

   maxavailable = (hmax - hmin) * capacity;
   minavailable = maxavailable;
   totalenergy = computeTotalEnergy(durations, demands, nvars);

   /* check if the smallest interval has a size such that the total energy fits, if so we can skip the propagator */
   if( (lcts[0] - ests[nvars-1]) * capacity >= totalenergy )
      return SCIP_OKAY;

   nests = nvars;

   /* loop over all variable in non-increasing order w.r.t. the latest completion time; thereby, the latest completion
    * times define the end of the time interval under investigation
    */
   for( v = nvars-1; v >= 0  && !(*cutoff); --v )
   {
      int flexenergy;
      int minbegin;
      int lbenergy;
      int lbcand;
      int i;

      lct = lcts[v];

      /* if the latest completion time is larger then hmax an infeasibility cannot be detected, since after hmax an
       * infinity capacity is available; hence we skip that
       */
      if( lct > hmax )
         continue;

      /* if the latest completion time is smaller then hmin we have to stop */
      if( lct <= hmin )
      {
         assert(v == 0 || lcts[v-1] <= lcts[v]);
         break;
      }

      /* if the latest completion time equals to previous end time, we can continue since this particular interval
       * induced by end was just analyzed
       */
      if( lct == end )
         continue;

      assert(lct < end);

      /* In case we only want to detect an overload (meaning no bound propagation) we can skip the interval; this is
       * the case if the free energy (the energy which is not occupied by any core) is smaller than the previous minimum
       * free energy; if so it means that in the next iterate the free-energy cannot be negative
       */
      if( !conshdlrdata->ttefinfer && end <= hmax && minavailable < maxavailable )
      {
         int freeenergy;

         assert(coreEnergyAfterLct[v] >= coreEnergyAfterEnd);
         assert(coreEnergyAfterEnd >= 0);

         /* compute the energy which is not consumed by the cores with in the interval [lct, end) */
         freeenergy = capacity * (end - lct) - coreEnergyAfterLct[v] + coreEnergyAfterEnd;

         if( freeenergy <= minavailable )
         {
            SCIPdebugMessage("skip latest completion time  <%d> (minimum available energy <%d>, free energy <%d>)\n", lct, minavailable, freeenergy);
            continue;
         }
      }

      SCIPdebugMessage("check intervals ending with <%d>\n", lct);

      end = lct;
      coreEnergyAfterEnd = coreEnergyAfterLct[v];

      flexenergy = 0;
      minavailable = maxavailable;
      minbegin = hmax;
      lbcand = -1;
      lbenergy = 0;

      /* loop over the job in non-increasing order w.r.t. the earliest start time; these earliest start time are
       * defining the beginning of the time interval under investigation; Thereby, the time interval gets wider and
       * wider
       */
      for( i = nests-1; i >= 0; --i )
      {
         SCIP_VAR* var;
         int freeenergy;
         int duration;
         int demand;
         int begin;
         int idx;
         int lst;

         idx = perm[i];
         assert(idx >= 0);
         assert(idx < nvars);
         assert(!(*cutoff));

         /* the earliest start time of the job */
         est = ests[i];

         /* if the job starts after the current end, we can skip it and do not need to consider it again since the
          * latest completion times (which define end) are scant in non-increasing order
          */
         if( end <= est )
         {
            nests--;
            continue;
         }

         /* check if the interval has a size such that the total energy fits, if so we can skip all intervals with the
          * current ending time
          */
         if( (end - est) * capacity >= totalenergy )
            break;

         var = vars[idx];
         assert(var != NULL);

         duration = durations[idx];
         assert(duration > 0);

         demand = demands[idx];
         assert(demand > 0);

         lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration;

         /* the latest start time of the free part of the job */
         lst = lsts[idx];

         /* in case the earliest start time is equal to minbegin, the job lies completely within the time window under
          * investigation; hence the overload check will do the the job
          */
         assert(est <= minbegin);
         if( minavailable < maxavailable && est < minbegin )
         {
            assert(!(*cutoff));

            /* try to tighten the upper bound */
            SCIP_CALL( tightenUbTTEF(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
                  var, duration, demand, est, lst, lct, minbegin, end, minavailable, &(newubs[idx]), &(ubinferinfos[idx]),
                  initialized, explanation, cutoff) );

            if( *cutoff )
               break;
         }

         SCIPdebugMessage("check variable <%s>[%g,%g] (duration %d, demands %d, est <%d>, lst of free part <%d>\n",
            SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var), duration, demand, est, lst);

         begin = est;
         assert(SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var)) == est);

         /* if the earliest start time is smaller than hmin we can stop here since the next job will not decrease the
          * free energy
          */
         if( begin < hmin )
            break;

         /* compute the contribution to the flexible energy */
         if( lct <= end )
         {
            /* if the jobs has to finish before the end, all the energy has to be scheduled */
            assert(lst >= begin);
            assert(flexenergies[idx] >= 0);
            flexenergy += flexenergies[idx];
         }
         else
         {
            /* the job partly overlaps with the end */
            int candenergy;
            int energy;

            /* compute the flexible energy which is part of the time interval for sure if the job is scheduled
             * w.r.t. latest start time
             *
             * @note we need to be aware of the effective horizon
             */
            energy = MIN(flexenergies[idx], demands[idx] * MAX(0, (end - lst)));
            assert(end - lst < duration);
            assert(energy >= 0);

            /* adjust the flexible energy of the time interval */
            flexenergy += energy;

            /* compute the flexible energy of the job which is not part of flexible energy of the time interval */
            candenergy = MIN(flexenergies[idx], demands[idx] * (end - begin)) - energy;
            assert(candenergy >= 0);

            /* check if we found a better candidate */
            if( candenergy > lbenergy )
            {
               lbenergy = candenergy;
               lbcand = idx;
            }
         }

         SCIPdebugMessage("time window [%d,%d) flexible energy <%d>\n", begin, end, flexenergy);
         assert(coreEnergyAfterEst[i] >= coreEnergyAfterEnd);

         /* compute the energy which is not used yet */
         freeenergy = capacity * (end - begin) - flexenergy - coreEnergyAfterEst[i] + coreEnergyAfterEnd;

         /* check overload */
         if( freeenergy < 0 )
         {
            SCIPdebugMessage("analyze overload  within time window [%d,%d) capacity %d\n", begin, end, capacity);

            /* initialize conflict analysis if conflict analysis is applicable */
            if( SCIPisConflictAnalysisApplicable(scip) )
            {
               /* analyze infeasibilty */
               SCIP_CALL( SCIPinitConflictAnalysis(scip) );

               SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
                     begin, end, NULL, SCIP_BOUNDTYPE_UPPER, NULL, SCIP_UNKNOWN,
                     conshdlrdata->usebdwidening, explanation) );

               (*initialized) = TRUE;
            }

            (*cutoff) = TRUE;

            /* for the statistic we count the number of times a cutoff was detected due the time-time-edge-finding */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffoverloadTTEF++ );

            break;
         }

         /* check if the available energy is not sufficent to schedule the flexible energy of the best candidate job */
         if( lbenergy > 0 && freeenergy < lbenergy )
         {
            int energy;
            int newlb;
            int ect;

            ect = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[lbcand])) + durations[lbcand];
            lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[lbcand]));

            /* remove the energy of our job from the ... */
            energy = freeenergy + (computeCoreWithInterval(begin, end, ect, lst) + MAX(0, end - lsts[lbcand])) * demands[lbcand];

            newlb = end - (int)(energy / demands[lbcand]);

            if( newlb > lst )
            {
               /* initialize conflict analysis if conflict analysis is applicable */
               if( SCIPisConflictAnalysisApplicable(scip) )
               {
                  SCIP_Real relaxedbd;

                  /* analyze infeasibilty */
                  SCIP_CALL( SCIPinitConflictAnalysis(scip) );

                  relaxedbd = lst + 1.0;

                  /* added to upper bound (which was overcut be new lower bound) of the variable */
                  SCIP_CALL( SCIPaddConflictUb(scip, vars[lbcand], NULL) );

                  SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
                        begin, end, vars[lbcand], SCIP_BOUNDTYPE_LOWER, NULL, relaxedbd,
                        conshdlrdata->usebdwidening, explanation) );

                  (*initialized) = TRUE;
               }

               (*cutoff) = TRUE;
               break;
            }
            else if( newlb > newlbs[lbcand] )
            {
               INFERINFO inferinfo;

               /* construct inference information */
               inferinfo = getInferInfo(PROPRULE_3_TTEF, begin, end);

               /* buffer upper bound change */
               lbinferinfos[lbcand] = inferInfoToInt(inferinfo);
               newlbs[lbcand] = newlb;
            }
         }

         /* check if the current interval has a smaller free energy */
         if( minavailable > freeenergy )
         {
            minavailable = freeenergy;
            minbegin = begin;
         }
         assert(minavailable >= 0);
      }
   }

   return SCIP_OKAY;
}

/** propagate the lower bounds and "opportunistically" the upper bounds using the time-table edge-finding algorithm */
static
SCIP_RETCODE propagateLbTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   int*                  newlbs,             /**< array to buffer new lower bounds */
   int*                  newubs,             /**< array to buffer new upper bounds */
   int*                  lbinferinfos,       /**< array to store the inference information for the lower bound changes */
   int*                  ubinferinfos,       /**< array to store the inference information for the upper bound changes */
   int*                  ects,               /**< array of earliest completion time of the flexible part in the same order as the variables */
   int*                  flexenergies,       /**< array of flexible energies in the same order as the variables */
   int*                  perm,               /**< permutation of the variables w.r.t. the non-decreasing order of the latest completion times */
   int*                  ests,               /**< array with earliest strart times sorted in non-decreasing order */
   int*                  lcts,               /**< array with latest completion times sorted in non-decreasing order */
   int*                  coreEnergyAfterEst, /**< core energy after the earliest start times */
   int*                  coreEnergyAfterLct, /**< core energy after the latest completion times */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int coreEnergyAfterStart;
   int maxavailable;
   int minavailable;
   int totalenergy;
   int nlcts;
   int begin;
   int minest;
   int maxlct;
   int start;
   int end;
   int v;

   if( *cutoff )
      return SCIP_OKAY;

   begin = hmin - 1;
   coreEnergyAfterStart = -1;

   minest = INT_MAX;
   maxlct = INT_MIN;

   /* compute earliest start and latest completion time of all jobs */
   for( v = 0; v < nvars; ++v )
   {
      start = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[v]));
      end = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[v])) + durations[v];

      minest = MIN(minest, start);
      maxlct = MAX(maxlct, end);
   }

   /* adjust the effective time horizon */
   hmin = MAX(hmin, minest);
   hmax = MIN(hmax, maxlct);

   maxavailable = (hmax - hmin) * capacity;
   minavailable = maxavailable;
   totalenergy = computeTotalEnergy(durations, demands, nvars);

   /* check if the smallest interval has a size such that the total energy fits, if so we can skip the propagator */
   if( (lcts[0] - ests[nvars-1]) * capacity >= totalenergy )
      return SCIP_OKAY;

   nlcts = 0;

   /* loop over all variable in non-decreasing order w.r.t. the earliest start times; thereby, the earliest start times
    * define the start of the time interval under investigation
    */
   for( v = 0; v < nvars; ++v )
   {
      int flexenergy;
      int minend;
      int ubenergy;
      int ubcand;
      int est;
      int i;

      est = ests[v];

      /* if the earliest start time is smaller then hmin an infeasibility cannot be detected, since before hmin an
       * infinity capacity is available; hence we skip that
       */
      if( est < hmin )
         continue;

      /* if the earliest start time is larger or equal then hmax we have to stop */
      if( est >= hmax )
         break;

      /* if the latest earliest start time equals to previous start time, we can continue since this particular interval
       * induced by start was just analyzed
       */
      if( est == begin )
         continue;

      assert(est > begin);

      SCIPdebugMessage("check intervals starting with <%d>\n", est);

      begin = est;
      coreEnergyAfterStart = coreEnergyAfterEst[v];

      flexenergy = 0;
      minavailable = maxavailable;
      minend = hmin;
      ubcand = -1;
      ubenergy = 0;

      /* loop over the job in non-decreasing order w.r.t. the latest completion time; these latest completion times are
       * defining the ending of the time interval under investigation; thereby, the time interval gets wider and wider
       */
      for( i = nlcts; i < nvars; ++i )
      {
         SCIP_VAR* var;
         int freeenergy;
         int duration;
         int demand;
         int idx;
         int lct;
         int ect;

         idx = perm[i];
         assert(idx >= 0);
         assert(idx < nvars);
         assert(!(*cutoff));

         /* the earliest start time of the job */
         lct = lcts[i];

         /* if the job has a latest completion time before the the current start, we can skip it and do not need to
          * consider it again since the earliest start times (which define the start) are scant in non-decreasing order
          */
         if( lct <= begin )
         {
            nlcts++;
            continue;
         }

         /* check if the interval has a size such that the total energy fits, if so we can skip all intervals which
          * start with current beginning time
          */
         if( (lct - begin) * capacity >= totalenergy )
            break;

         var = vars[idx];
         assert(var != NULL);

         duration = durations[idx];
         assert(duration > 0);

         demand = demands[idx];
         assert(demand > 0);

         est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));

         /* the earliest completion time of the flexible part of the job */
         ect = ects[idx];

         /* in case the latest completion time is equal to minend, the job lies completely within the time window under
          * investigation; hence the overload check will do the the job
          */
         assert(lct >= minend);
         if( minavailable < maxavailable && lct > minend )
         {
            assert(!(*cutoff));

            /* try to tighten the upper bound */
            SCIP_CALL( tightenLbTTEF(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
                  var, duration, demand, est, ect, lct, begin, minend, minavailable, &(newlbs[idx]), &(lbinferinfos[idx]),
                  initialized, explanation, cutoff) );

            if( *cutoff )
               return SCIP_OKAY;
         }

         SCIPdebugMessage("check variable <%s>[%g,%g] (duration %d, demands %d, est <%d>, ect of free part <%d>\n",
            SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var), duration, demand, est, ect);

         end = lct;
         assert(SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration == lct);

         /* if the latest completion time is larger than hmax we can stop here since the next job will not decrease the
          * free energy
          */
         if( end > hmax )
            break;

         /* compute the contribution to the flexible energy */
         if( est >= begin )
         {
            /* if the jobs has to finish before the end, all the energy has to be scheduled */
            assert(ect <= end);
            assert(flexenergies[idx] >= 0);
            flexenergy += flexenergies[idx];
         }
         else
         {
            /* the job partly overlaps with the end */
            int candenergy;
            int energy;

            /* compute the flexible energy which is part of the time interval for sure if the job is scheduled
             * w.r.t. latest start time
             *
             * @note we need to be aware of the effective horizon
             */
            energy = MIN(flexenergies[idx], demands[idx] * MAX(0, (ect - begin)));
            assert(ect - begin < duration);
            assert(energy >= 0);

            /* adjust the flexible energy of the time interval */
            flexenergy += energy;

            /* compute the flexible energy of the job which is not part of flexible energy of the time interval */
            candenergy = MIN(flexenergies[idx], demands[idx] * (end - begin)) - energy;
            assert(candenergy >= 0);

            /* check if we found a better candidate */
            if( candenergy > ubenergy )
            {
               ubenergy = candenergy;
               ubcand = idx;
            }
         }

         SCIPdebugMessage("time window [%d,%d) flexible energy <%d>\n", begin, end, flexenergy);
         assert(coreEnergyAfterLct[i] <= coreEnergyAfterStart);

         /* compute the energy which is not used yet */
         freeenergy = capacity * (end - begin) - flexenergy - coreEnergyAfterStart + coreEnergyAfterLct[i];

         /* check overload */
         if( freeenergy < 0 )
         {
            SCIPdebugMessage("analyze overload  within time window [%d,%d) capacity %d\n", begin, end, capacity);

            /* initialize conflict analysis if conflict analysis is applicable */
            if( SCIPisConflictAnalysisApplicable(scip) )
            {
               /* analyze infeasibilty */
               SCIP_CALL( SCIPinitConflictAnalysis(scip) );

               SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
                     begin, end, NULL, SCIP_BOUNDTYPE_UPPER, NULL, SCIP_UNKNOWN,
                     conshdlrdata->usebdwidening, explanation) );

               (*initialized) = TRUE;
            }

            (*cutoff) = TRUE;

            /* for the statistic we count the number of times a cutoff was detected due the time-time-edge-finding */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffoverloadTTEF++ );

            return SCIP_OKAY;
         }

         /* check if the available energy is not sufficent to schedule the flexible energy of the best candidate job */
         if( ubenergy > 0 && freeenergy < ubenergy )
         {
            int energy;
            int newub;
            int lst;

            duration = durations[ubcand];
            assert(duration > 0);

            ect = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[ubcand])) + duration;
            lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[ubcand]));

            /* remove the energy of our job from the ... */
            energy = freeenergy + (computeCoreWithInterval(begin, end, ect, lst) + MAX(0, ects[ubcand] - begin)) * demands[ubcand];

            newub = begin - duration + (int)(energy / demands[ubcand]);

            if( newub < ect - duration )
            {
               /* initialize conflict analysis if conflict analysis is applicable */
               if( SCIPisConflictAnalysisApplicable(scip) )
               {
                  SCIP_Real relaxedbd;
                  /* analyze infeasibilty */
                  SCIP_CALL( SCIPinitConflictAnalysis(scip) );

                  relaxedbd = ect - duration - 1.0;

                  /* added to lower bound (which was undercut be new upper bound) of the variable */
                  SCIP_CALL( SCIPaddConflictUb(scip, vars[ubcand], NULL) );

                  SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
                        begin, end, vars[ubcand], SCIP_BOUNDTYPE_UPPER, NULL, relaxedbd,
                        conshdlrdata->usebdwidening, explanation) );

                  (*initialized) = TRUE;
               }

               (*cutoff) = TRUE;
               return SCIP_OKAY;
            }
            else if( newub < newubs[ubcand] )
            {
               INFERINFO inferinfo;

               /* construct inference information */
               inferinfo = getInferInfo(PROPRULE_3_TTEF, begin, end);

               /* buffer upper bound change */
               ubinferinfos[ubcand] = inferInfoToInt(inferinfo);
               newubs[ubcand] = newub;
            }
         }

         /* check if the current interval has a smaller free energy */
         if( minavailable > freeenergy )
         {
            minavailable = freeenergy;
            minend = end;
         }
         assert(minavailable >= 0);
      }
   }

   return SCIP_OKAY;
}

/** checks whether the instance is infeasible due to a overload within a certain time frame using the idea of time-table
 *  edge-finding
 *
 *  @note The algorithm is based on the following two papers:
 *        - Petr Vilim, "Timetable Edge Finding Filtering Algorithm for Discrete Cumulative Resources", In: Tobias
 *          Achterberg and J. Christopher Beck (Eds.), Integration of AI and OR Techniques in Constraint Programming for
 *          Combinatorial Optimization Problems (CPAIOR 2011), LNCS 6697, pp 230--245
 *        - Andreas Schutt, Thibaut Feydy, and Peter J. Stuckey, "Explaining Time-Table-Edge-Finding Propagation for the
 *          Cumulative Resource Constraint (submitted to CPAIOR 2013)
 */
static
SCIP_RETCODE propagateTTEF(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PROFILE*         profile,            /**< current core profile */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated (needed to SCIPinferVar**Cons()) */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int* coreEnergyAfterEst;
   int* coreEnergyAfterLct;
   int* flexenergies;
   int* permests;
   int* permlcts;
   int* lcts;
   int* ests;
   int* ects;
   int* lsts;

   int* newlbs;
   int* newubs;
   int* lbinferinfos;
   int* ubinferinfos;

   int v;

   /* check if a cutoff was already detected */
   if( (*cutoff) )
      return SCIP_OKAY;

   /* check if at least the basic overload checking should be perfomed */
   if( !conshdlrdata->ttefcheck )
      return SCIP_OKAY;

   SCIPdebugMessage("run time-table edge-finding overload checking\n");

   SCIP_CALL( SCIPallocBufferArray(scip, &coreEnergyAfterEst, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &coreEnergyAfterLct, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &flexenergies, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &permlcts, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &permests, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &lcts, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &ests, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &ects, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &lsts, nvars) );

   SCIP_CALL( SCIPallocBufferArray(scip, &newlbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &newubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &lbinferinfos, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &ubinferinfos, nvars) );

   /* we need to buffer the bound changes since the propagation algorithm cannot handle new bound dynamically */
   for( v = 0; v < nvars; ++v )
   {
      newlbs[v] = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[v]));
      newubs[v] = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[v]));
      lbinferinfos[v] = 0;
      ubinferinfos[v] = 0;
   }

   /* collect earliest start times, latest completion time, and free energy contributions */
   collectDataTTEF(scip, nvars, vars, durations, demands, hmin, hmax, permests, ests, permlcts, lcts, ects, lsts, flexenergies);

   /* sort the earliest start times and latest completion in non-decreasing order */
   SCIPsortIntInt(ests, permests, nvars);
   SCIPsortIntInt(lcts, permlcts, nvars);

   /* compute for the different earliest start and latest completion time the core energy of the corresponding time
    * points
    */
   SCIP_CALL( computeCoreEngeryAfter(scip, profile, nvars, ests, lcts, coreEnergyAfterEst, coreEnergyAfterLct) );

   /* propagate the upper bounds and "opportunistically" the lower bounds */
   SCIP_CALL( propagateUbTTEF(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
         newlbs, newubs, lbinferinfos, ubinferinfos, lsts, flexenergies,
         permests, ests, lcts, coreEnergyAfterEst, coreEnergyAfterLct, initialized, explanation, cutoff) );

   /* propagate the lower bounds and "opportunistically" the upper bounds */
   SCIP_CALL( propagateLbTTEF(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
         newlbs, newubs, lbinferinfos, ubinferinfos, ects, flexenergies,
         permlcts, ests, lcts, coreEnergyAfterEst, coreEnergyAfterLct, initialized, explanation, cutoff) );

   /* apply the buffer bound changes */
   for( v = 0; v < nvars && !(*cutoff); ++v )
   {
      SCIP_Bool infeasible;
      SCIP_Bool tightened;

      SCIP_CALL( SCIPinferVarLbCons(scip, vars[v], (SCIP_Real)newlbs[v], cons, lbinferinfos[v], TRUE, &infeasible, &tightened) );

      /* since we change first the lower bound of the variable an infeasibilty should be detected */
      assert(!infeasible);

      if( tightened )
      {
         (*nchgbds)++;

         /* for the statistic we count the number of times a cutoff was detected due the time-time */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nlbTTEF++ );
      }


      SCIP_CALL( SCIPinferVarUbCons(scip, vars[v], (SCIP_Real)newubs[v], cons, ubinferinfos[v], TRUE, &infeasible, &tightened) );

      /* since upper bound was compute w.r.t. the "old" bound the previous lower bound update together with this upper
       * bound update can be infeasible
       */
      if( infeasible )
      {
         if( SCIPisConflictAnalysisApplicable(scip) )
         {
            INFERINFO inferinfo;
            SCIP_VAR* var;
            int begin;
            int end;

            var = vars[v];
            assert(var != NULL);

            /* initialize conflict analysis */
            SCIP_CALL( SCIPinitConflictAnalysis(scip) );

            /* convert int to inference information */
            inferinfo = intToInferInfo(ubinferinfos[v]);

            /* collect time window from inference information */
            begin = inferInfoGetData1(inferinfo);
            end = inferInfoGetData2(inferinfo);
            assert(begin < end);

            /* added to lower bound (which was undercut be new upper bound) of the variable */
            SCIP_CALL( SCIPaddConflictLb(scip, var, NULL) );

            /* analysis the upper bound change */
            SCIP_CALL( analyzeEnergyRequirement(scip, nvars, vars, durations, demands, capacity,
                  begin, end, var, SCIP_BOUNDTYPE_UPPER, NULL, SCIPvarGetLbLocal(vars[v]) - 1.0,
                  conshdlrdata->usebdwidening, explanation) );

            (*initialized) = TRUE;
         }

         /* for the statistic we count the number of times a cutoff was detected due the time-time */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffoverloadTTEF++ );

         (*cutoff) = TRUE;
         break;
      }

      if( tightened )
      {
         (*nchgbds)++;

         /* for the statistic we count the number of times a cutoff was detected due the time-time */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nubTTEF++ );
      }
   }

   SCIPfreeBufferArray(scip, &ubinferinfos);
   SCIPfreeBufferArray(scip, &lbinferinfos);
   SCIPfreeBufferArray(scip, &newubs);
   SCIPfreeBufferArray(scip, &newlbs);

   /* free buffer arrays */
   SCIPfreeBufferArray(scip, &lsts);
   SCIPfreeBufferArray(scip, &ects);
   SCIPfreeBufferArray(scip, &ests);
   SCIPfreeBufferArray(scip, &lcts);
   SCIPfreeBufferArray(scip, &permests);
   SCIPfreeBufferArray(scip, &permlcts);
   SCIPfreeBufferArray(scip, &flexenergies);
   SCIPfreeBufferArray(scip, &coreEnergyAfterLct);
   SCIPfreeBufferArray(scip, &coreEnergyAfterEst);

   return SCIP_OKAY;
}

/** a cumulative condition is not satisfied if its capacity is exceeded at a time where jobs cannot be shifted (core)
 *  anymore we build up a cumulative profile of all cores of jobs and try to improve bounds of all jobs; also known as
 *  time table propagator
 */
static
SCIP_RETCODE propagateTimetable(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PROFILE*         profile,            /**< core profile */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated (needed to SCIPinferVar**Cons()) */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_Bool infeasible;
   int v;

   assert(scip != NULL);
   assert(nvars > 0);
   assert(cons != NULL);
   assert(cutoff !=  NULL);

   /* check if already a cutoff was detected */
   if( (*cutoff) )
      return SCIP_OKAY;

   /* check if the time tabling should infer bounds */
   if( !conshdlrdata->ttinfer )
      return SCIP_OKAY;

   assert(*initialized ==  FALSE);

   SCIPdebugMessage("propagate cores of cumulative condition of constraint <%s>[%d,%d) <= %d\n",
      SCIPconsGetName(cons), hmin, hmax, capacity);

   infeasible = FALSE;

   /* if core profile is empty; nothing to do */
   if( SCIPprofileGetNTimepoints(profile) <= 1 )
      return SCIP_OKAY;

   /* start checking each job whether the bounds can be improved */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      int demand;
      int duration;
      int begin;
      int end;
      int est;
      int lst;

      var = vars[v];
      assert(var != NULL);

      duration = durations[v];
      assert(duration > 0);

      /* collect earliest and latest start time */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* check if the start time variables is already fixed; in that case we can ignore the job */
      if( est == lst )
         continue;

      /* check if the job runs completely outside of the effective horizon [hmin, hmax); if so skip it */
      if( lst + duration <= hmin || est >= hmax )
         continue;

      /* compute core interval w.r.t. effective time horizon */
      begin = MAX(hmin, lst);
      end = MIN(hmax, est + duration);

      demand = demands[v];
      assert(demand > 0);

      /* if the job has a core, remove it first */
      if( begin < end  )
      {
         SCIPdebugMessage("variable <%s>[%g,%g] (duration %d, demand %d): remove core [%d,%d)\n",
            SCIPvarGetName(var), SCIPvarGetLbLocal(var), SCIPvarGetUbLocal(var), duration, demand, begin, end);

         SCIP_CALL( SCIPprofileDeleteCore(profile, begin, end, demand) );
      }

      /* first try to update the earliest start time */
      SCIP_CALL( coretimesUpdateLb(scip, nvars, vars, durations, demands, capacity, hmin, hmax, cons,
            profile, v, nchgbds, conshdlrdata->usebdwidening, initialized, explanation, cutoff) );

      if( *cutoff )
         break;

      /* second try to update the latest start time */
      SCIP_CALL( coretimesUpdateUb(scip, var, duration, demand, capacity, cons,
            profile, v, nchgbds) );

      if( *cutoff )
         break;

      /* collect the potentially updated earliest and latest start time */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* compute core interval w.r.t. effective time horizon */
      begin = MAX(hmin, lst);
      end = MIN(hmax, est + duration);

      /* after updating the bound we might have a new core */
      if( begin < end )
      {
         int pos;

         SCIPdebugMessage("variable <%s>[%d,%d] (duration %d, demand %d): add core [%d,%d)\n",
            SCIPvarGetName(var), est, lst, duration, demand, begin, end);

         SCIP_CALL( SCIPprofileInsertCore(profile, begin, end, demand, &pos, &infeasible) );

         if( infeasible )
         {
            /* use conflict analysis to analysis the core insertion which was infeasible */
            SCIP_CALL( analyseInfeasibelCoreInsertion(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
                  var, duration, demand, SCIPprofileGetTime(profile, pos), conshdlrdata->usebdwidening, initialized, explanation) );

            if( explanation != NULL )
               explanation[v] = TRUE;

            (*cutoff) = TRUE;

            /* for the statistic we count the number of times a cutoff was detected due the time-time */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutofftimetable++ );

            break;
         }
      }
   }

   return SCIP_OKAY;
}


/** node data structure for the binary tree used for edgefinding (with overload checking) */
struct SCIP_NodeData
{
   SCIP_VAR*             var;                /**< start time variable of the job if the node data belongs to a leaf, otherwise NULL */
   SCIP_Real             key;                /**< key which is to insert the corresponding search node */
   int                   est;                /**< earliest start time if the node data belongs to a leaf */
   int                   lct;                /**< latest completion time if the node data belongs to a leaf */
   int                   demand;             /**< demand of the job if the node data belongs to a leaf */
   int                   duration;           /**< duration of the job if the node data belongs to a leaf */
   int                   leftadjust;         /**< left adjustments of the duration w.r.t. hmin */
   int                   rightadjust;        /**< right adjustments of the duration w.r.t. hmax */
   int                   enveloptheta;       /**< the maximal energy of a subset of jobs part of the theta set */
   int                   energytheta;        /**< energy of the subset of the jobs which are part of theta set */
   int                   energylambda;
   int                   enveloplambda;
   int                   idx;                /**< index of the start time variable in the (global) variable array */
   SCIP_Bool             intheta;            /**< belongs the node to the theta set (otherwise to the lambda set) */
};
typedef struct SCIP_NodeData SCIP_NODEDATA;

/** creates a node data structure */
static
SCIP_RETCODE createNodedata(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_NODEDATA**       nodedata            /**< pointer to store the create node data */
   )
{
   SCIP_CALL( SCIPallocBuffer(scip, nodedata) );
   (*nodedata)->var = NULL;
   (*nodedata)->key = SCIP_INVALID;
   (*nodedata)->est = INT_MIN;
   (*nodedata)->lct = INT_MAX;
   (*nodedata)->duration = 0;
   (*nodedata)->demand = 0;
   (*nodedata)->enveloptheta = -1;
   (*nodedata)->energytheta = 0;
   (*nodedata)->enveloplambda = -1;
   (*nodedata)->energylambda = -1;
   (*nodedata)->idx = -1;
   (*nodedata)->intheta = TRUE;

   return SCIP_OKAY;
}

/** frees a  node data structure */
static
void freeNodedata(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_NODEDATA**       nodedata            /**< pointer to store node data which should be freed */
   )
{
   if( *nodedata != NULL )
   {
      SCIPfreeBuffer(scip, nodedata);
   }
}

/** update node data structure strating form the given node along the path to the root node */
static
SCIP_RETCODE updateEnvelop(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_BTNODE*          node                /**< search node which inserted */
   )
{
   SCIP_BTNODE* left;
   SCIP_BTNODE* right;
   SCIP_NODEDATA* nodedata;
   SCIP_NODEDATA* leftdata;
   SCIP_NODEDATA* rightdata;

   SCIPdebugMessage("update envelop starting from node <%p>\n", (void*)node);

   if( SCIPbtnodeIsLeaf(node) )
      node = SCIPbtnodeGetParent(node);

   while( node != NULL )
   {
      /* get node data */
      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
      assert(nodedata != NULL);

      /* collect node data from left node */
      left = SCIPbtnodeGetLeftchild(node);
      assert(left != NULL);
      leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
      assert(leftdata != NULL);

      /* collect node data from right node */
      right = SCIPbtnodeGetRightchild(node);
      assert(right != NULL);
      rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
      assert(rightdata != NULL);

      /* update envelop and energy */
      if( leftdata->enveloptheta >= 0 )
      {
         assert(rightdata->energytheta != -1);
         nodedata->enveloptheta = MAX(leftdata->enveloptheta + rightdata->energytheta, rightdata->enveloptheta);
      }
      else
         nodedata->enveloptheta = rightdata->enveloptheta;

      assert(leftdata->energytheta != -1);
      assert(rightdata->energytheta != -1);
      nodedata->energytheta = leftdata->energytheta + rightdata->energytheta;

      if( leftdata->enveloplambda >= 0 )
      {
         assert(rightdata->energytheta != -1);
         nodedata->enveloplambda = MAX(leftdata->enveloplambda + rightdata->energytheta, rightdata->enveloplambda);
      }
      else
         nodedata->enveloplambda = rightdata->enveloplambda;

      if( leftdata->enveloptheta >= 0 && rightdata->energylambda >= 0 )
         nodedata->enveloplambda = MAX(nodedata->enveloplambda, leftdata->enveloptheta + rightdata->energylambda);

      SCIPdebugMessage("node <%p> lambda envelop %d\n", (void*)node, nodedata->enveloplambda);

      if( leftdata->energylambda >= 0 && rightdata->energylambda >= 0 )
      {
         assert(rightdata->energytheta != -1);
         assert(leftdata->energytheta != -1);
         nodedata->energylambda = MAX(leftdata->energylambda + rightdata->energytheta, leftdata->energytheta + rightdata->energylambda);
      }
      else if( rightdata->energylambda >= 0 )
      {
         assert(leftdata->energytheta != -1);
         nodedata->energylambda = leftdata->energytheta + rightdata->energylambda;
      }
      else if( leftdata->energylambda >= 0 )
      {
         assert(rightdata->energytheta != -1);
         nodedata->energylambda = leftdata->energylambda + rightdata->energytheta;
      }
      else
         nodedata->energylambda = -1;

      /* go to parent */
      node = SCIPbtnodeGetParent(node);
   }

   SCIPdebugMessage("updating done\n");

   return SCIP_OKAY;
}

/** updates the key of the first parent on the trace which comes from left */
static
void updateKeyOnTrace(
   SCIP_BTNODE*          node,               /**< node to start the trace */
   SCIP_Real             key                 /**< update search key */
   )
{
   assert(node != NULL);

   while( !SCIPbtnodeIsRoot(node) )
   {
      SCIP_BTNODE* parent;

      parent = SCIPbtnodeGetParent(node);
      assert(parent != NULL);

      if( SCIPbtnodeIsLeftchild(node) )
      {
         SCIP_NODEDATA* nodedata;

         nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(parent);
         assert(nodedata != NULL);

         nodedata->key = key;
         return;
      }

      node = parent;
   }
}


/** deletes the given node and updates all envelops */
static
SCIP_RETCODE deleteLambdaLeaf(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_BT*              tree,               /**< binary tree */
   SCIP_BTNODE*          node                /**< node to be deleted */
   )
{
   SCIP_BTNODE* parent;
   SCIP_BTNODE* grandparent;
   SCIP_BTNODE* sibling;

   assert(scip != NULL);
   assert(tree != NULL);
   assert(node != NULL);

   assert(SCIPbtnodeIsLeaf(node));
   assert(!SCIPbtnodeIsRoot(node));

   SCIPdebugMessage("delete node <%p>\n", (void*)node);

   parent = SCIPbtnodeGetParent(node);
   assert(parent != NULL);
   if( SCIPbtnodeIsLeftchild(node) )
   {
      sibling = SCIPbtnodeGetRightchild(parent);
      SCIPbtnodeSetRightchild(parent, NULL);
   }
   else
   {
      sibling = SCIPbtnodeGetLeftchild(parent);
      SCIPbtnodeSetLeftchild(parent, NULL);
   }
   assert(sibling != NULL);

   grandparent = SCIPbtnodeGetParent(parent);

   if( grandparent != NULL )
   {
      /* reset parent of sibling */
      SCIPbtnodeSetParent(sibling, grandparent);

      /* reset child of grandparent to sibling */
      if( SCIPbtnodeIsLeftchild(parent) )
      {
         SCIPbtnodeSetLeftchild(grandparent, sibling);
      }
      else
      {
         SCIP_NODEDATA* nodedata;

         assert(SCIPbtnodeIsRightchild(parent));
         SCIPbtnodeSetRightchild(grandparent, sibling);

         nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(sibling);

         updateKeyOnTrace(grandparent, nodedata->key);
      }

      SCIP_CALL( updateEnvelop(scip, grandparent) );
   }
   else
   {
      SCIPbtnodeSetParent(sibling, NULL);

      SCIPbtSetRoot(tree, sibling);
   }


   SCIPbtnodeFree(tree, &parent);

   return SCIP_OKAY;
}

/** moves a node form the theta set into the lambda set and updates the envelops */
static
SCIP_RETCODE moveNodeToLambda(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_BT*              tree,               /**< binary tree */
   SCIP_BTNODE*          node                /**< node to move into the lambda set */
   )
{
   SCIP_NODEDATA* nodedata;

   assert(scip != NULL);
   assert(tree != NULL);
   assert(node != NULL);

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);
   assert(nodedata->intheta);

   /* move the contributions form the theta set into the lambda set */
   assert(nodedata->enveloptheta != -1);
   assert(nodedata->energytheta != -1);
   assert(nodedata->enveloplambda == -1);
   assert(nodedata->energylambda == -1);
   nodedata->enveloplambda = nodedata->enveloptheta;
   nodedata->energylambda = nodedata->energytheta;

   nodedata->enveloptheta = -1;
   nodedata->energytheta = 0;
   nodedata->intheta = FALSE;

   /* update the energy and envelop values on trace */
   SCIP_CALL( updateEnvelop(scip, node) );

   return SCIP_OKAY;
}

/** inserts a node into the theta set and update the envelops */
static
SCIP_RETCODE insertThetanode(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_BT*              tree,               /**< binary tree */
   SCIP_BTNODE*          node,               /**< node to insert */
   SCIP_NODEDATA**       nodedatas,          /**< array of node data */
   int*                  nnodedatas          /**< pointer to number of node data */
   )
{
   /* if the tree is empty the node will be the root node */
   if( SCIPbtIsEmpty(tree) )
   {
      SCIPbtSetRoot(tree, node);
   }
   else
   {
      SCIP_NODEDATA* newnodedata;
      SCIP_NODEDATA* leafdata;
      SCIP_NODEDATA* nodedata;
      SCIP_BTNODE* leaf;
      SCIP_BTNODE* newnode;
      SCIP_BTNODE* parent;

      leaf = SCIPbtGetRoot(tree);
      assert(leaf != NULL);

      leafdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaf);
      assert(leafdata != NULL);

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
      assert(nodedata != NULL);
      assert(nodedata->intheta);

      /* find the position to insert the node */
      while( !SCIPbtnodeIsLeaf(leaf) )
      {
         if( nodedata->key < leafdata->key )
            leaf = SCIPbtnodeGetLeftchild(leaf);
         else
            leaf = SCIPbtnodeGetRightchild(leaf);

         leafdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaf);
         assert(leafdata != NULL);
      }

      assert(leaf != NULL);
      assert(leaf != node);

      /* create node data */
      SCIP_CALL( createNodedata(scip, &newnodedata) );

      /* create a new node */
      SCIP_CALL( SCIPbtnodeCreate(tree, &newnode, newnodedata) );
      assert(newnode != NULL);

      /* store node data to be able to delete them latter */
      nodedatas[*nnodedatas] = newnodedata;
      (*nnodedatas)++;

      parent = SCIPbtnodeGetParent(leaf);

      if( parent != NULL )
      {
         SCIPbtnodeSetParent(newnode, parent);

         /* check if the node is the left child */
         if( SCIPbtnodeGetLeftchild(parent) == leaf )
         {
            SCIPbtnodeSetLeftchild(parent, newnode);
         }
         else
         {
            SCIPbtnodeSetRightchild(parent, newnode);
         }
      }
      else
         SCIPbtSetRoot(tree, newnode);

      if( nodedata->key < leafdata->key )
      {
         /* node is on the left */
         SCIPbtnodeSetLeftchild(newnode, node);
         SCIPbtnodeSetRightchild(newnode, leaf);
         newnodedata->key = nodedata->key;
      }
      else
      {
         /* leaf is on the left */
         SCIPbtnodeSetLeftchild(newnode, leaf);
         SCIPbtnodeSetRightchild(newnode, node);
         newnodedata->key = leafdata->key;
      }

      SCIPbtnodeSetParent(leaf, newnode);
      SCIPbtnodeSetParent(node, newnode);
   }

   /* update envelop */
   SCIP_CALL( updateEnvelop(scip, node) );

   return SCIP_OKAY;
}

/** returns the leaf responsible for the lambda energy */
static
SCIP_BTNODE* findResponsibleLambdaLeafTraceEnergy(
   SCIP_BTNODE*          node                /**< node which defines the subtree beases on the lambda energy */
   )
{
   SCIP_BTNODE* left;
   SCIP_BTNODE* right;
   SCIP_NODEDATA* nodedata;
   SCIP_NODEDATA* leftdata;
   SCIP_NODEDATA* rightdata;

   assert(node != NULL);

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);

   /* check if the node is the (responsible) leaf */
   if( SCIPbtnodeIsLeaf(node) )
   {
      assert(!nodedata->intheta);
      return node;
   }

   left = SCIPbtnodeGetLeftchild(node);
   assert(left != NULL);

   leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
   assert(leftdata != NULL);

   right = SCIPbtnodeGetRightchild(node);
   assert(right != NULL);

   rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
   assert(rightdata != NULL);

   assert(nodedata->energylambda != -1);
   assert(rightdata->energytheta != -1);

   if( leftdata->energylambda >= 0 && nodedata->energylambda == leftdata->energylambda + rightdata->energytheta )
      return findResponsibleLambdaLeafTraceEnergy(left);

   assert(leftdata->energytheta != -1);
   assert(rightdata->energylambda != -1);
   assert(nodedata->energylambda == leftdata->energytheta + rightdata->energylambda);

   return findResponsibleLambdaLeafTraceEnergy(right);
}

/** returns the leaf responsible for the lambda envelop */
static
SCIP_BTNODE* findResponsibleLambdaLeafTraceEnvelop(
   SCIP_BTNODE*          node                /**< node which defines the subtree beases on the lambda envelop */
   )
{
   SCIP_BTNODE* left;
   SCIP_BTNODE* right;
   SCIP_NODEDATA* nodedata;
   SCIP_NODEDATA* leftdata;
   SCIP_NODEDATA* rightdata;

   assert(node != NULL);

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);

   /* check if the node is the (responsible) leaf */
   if( SCIPbtnodeIsLeaf(node) )
   {
      assert(!nodedata->intheta);
      return node;
   }

   left = SCIPbtnodeGetLeftchild(node);
   assert(left != NULL);

   leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
   assert(leftdata != NULL);

   right = SCIPbtnodeGetRightchild(node);
   assert(right != NULL);

   rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
   assert(rightdata != NULL);

   assert(nodedata->enveloplambda != -1);
   assert(rightdata->energytheta != -1);

   /* check if the left or right child is the one defining the envelop for the lambda set */
   if( leftdata->enveloplambda >= 0 && nodedata->enveloplambda == leftdata->enveloplambda + rightdata->energytheta )
      return findResponsibleLambdaLeafTraceEnvelop(left);
   else if( leftdata->enveloptheta >= 0 && rightdata->energylambda >= 0
      && nodedata->enveloplambda == leftdata->enveloptheta + rightdata->energylambda )
      return findResponsibleLambdaLeafTraceEnergy(right);

   assert(rightdata->enveloplambda != -1);
   assert(nodedata->enveloplambda == rightdata->enveloplambda);

   return findResponsibleLambdaLeafTraceEnvelop(right);
}


/** reports all elements from set theta to generate a conflicting set */
static
void collectThetaSubtree(
   SCIP_BTNODE*          node,               /**< node within a theta subtree */
   SCIP_BTNODE**         omegaset,           /**< array to store the collected jobs */
   int*                  nelements,          /**< pointer to store the number of elements in omegaset */
   int*                  est,                /**< pointer to store the earliest start time of the omega set */
   int*                  lct,                /**< pointer to store the latest start time of the omega set */
   int*                  energy              /**< pointer to store the energy of the omega set */
   )
{
   SCIP_NODEDATA* nodedata;

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);

   if( !SCIPbtnodeIsLeaf(node) )
   {
      collectThetaSubtree(SCIPbtnodeGetLeftchild(node), omegaset, nelements, est, lct, energy);
      collectThetaSubtree(SCIPbtnodeGetRightchild(node), omegaset, nelements, est, lct, energy);
   }
   else if( nodedata->intheta )
   {
      assert(nodedata->var != NULL);
      SCIPdebugMessage("add variable <%s> as elements %d to omegaset\n", SCIPvarGetName(nodedata->var), *nelements);

      omegaset[*nelements] = node;
      (*est) = MIN(*est, nodedata->est);
      (*lct) = MAX(*lct, nodedata->lct);
      (*energy) += (nodedata->duration - nodedata->leftadjust - nodedata->rightadjust) * nodedata->demand;
      (*nelements)++;
   }
}


/** collect the jobs (omega set) which are contribute to theta envelop from the theta set */
static
void traceThetaEnvelop(
   SCIP_BTNODE*          node,               /**< node whose theta envelop needs to be backtracked */
   SCIP_BTNODE**         omegaset,           /**< array to store the collected jobs */
   int*                  nelements,          /**< pointer to store the number of elements in omegaset */
   int*                  est,                /**< pointer to store the earliest start time of the omega set */
   int*                  lct,                /**< pointer to store the latest start time of the omega set */
   int*                  energy              /**< pointer to store the energy of the omega set */
   )
{
   assert(node != NULL);

   if( SCIPbtnodeIsLeaf(node) )
   {
      collectThetaSubtree(node, omegaset, nelements, est, lct, energy);
   }
   else
   {
      SCIP_BTNODE* left;
      SCIP_BTNODE* right;
      SCIP_NODEDATA* nodedata;
      SCIP_NODEDATA* leftdata;
      SCIP_NODEDATA* rightdata;

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
      assert(nodedata != NULL);


      left = SCIPbtnodeGetLeftchild(node);
      assert(left != NULL);

      leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
      assert(leftdata != NULL);

      right = SCIPbtnodeGetRightchild(node);
      assert(right != NULL);

      rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
      assert(rightdata != NULL);

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
      assert(nodedata != NULL);

      assert(nodedata->enveloptheta != -1);
      assert(rightdata->energytheta != -1);

      if( leftdata->enveloptheta >= 0 && nodedata->enveloptheta == leftdata->enveloptheta + rightdata->energytheta )
      {
         traceThetaEnvelop(left, omegaset, nelements, est, lct, energy);
         collectThetaSubtree(right, omegaset, nelements, est, lct, energy);
      }
      else
      {
         assert(rightdata->enveloptheta != -1);
         assert(nodedata->enveloptheta == rightdata->enveloptheta);
         traceThetaEnvelop(right, omegaset, nelements, est, lct, energy);
      }
   }
}

/** collect the jobs (omega set) which are contribute to lambda envelop from the theta set */
static
void traceLambdaEnergy(
   SCIP_BTNODE*          node,               /**< node whose lambda envelop needs to be backtracked */
   SCIP_BTNODE**         omegaset,           /**< array to store the collected jobs */
   int*                  nelements,          /**< pointer to store the number of elements in omega set */
   int*                  est,                /**< pointer to store the earliest start time of the omega set */
   int*                  lct,                /**< pointer to store the latest start time of the omega set */
   int*                  energy              /**< pointer to store the energy of the omega set */
   )
{
   SCIP_BTNODE* left;
   SCIP_BTNODE* right;
   SCIP_NODEDATA* nodedata;
   SCIP_NODEDATA* leftdata;
   SCIP_NODEDATA* rightdata;

   assert(node != NULL);

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);

   /* check if the node is a leaf */
   if( SCIPbtnodeIsLeaf(node) )
      return;

   left = SCIPbtnodeGetLeftchild(node);
   assert(left != NULL);

   leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
   assert(leftdata != NULL);

   right = SCIPbtnodeGetRightchild(node);
   assert(right != NULL);

   rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
   assert(rightdata != NULL);

   assert(nodedata->energylambda != -1);
   assert(rightdata->energytheta != -1);

   if( leftdata->energylambda >= 0 && nodedata->energylambda == leftdata->energylambda + rightdata->energytheta )
   {
      traceLambdaEnergy(left, omegaset, nelements, est, lct, energy);
      collectThetaSubtree(right, omegaset, nelements, est, lct, energy);
   }
   else
   {
      assert(leftdata->energytheta != -1);
      assert(rightdata->energylambda != -1);
      assert(nodedata->energylambda == leftdata->energytheta + rightdata->energylambda);

      collectThetaSubtree(left, omegaset, nelements, est, lct, energy);
      traceLambdaEnergy(right, omegaset, nelements, est, lct, energy);
   }
}

/** collect the jobs (omega set) which are contribute to lambda envelop from the theta set */
static
void traceLambdaEnvelop(
   SCIP_BTNODE*          node,               /**< node whose lambda envelop needs to be backtracked */
   SCIP_BTNODE**         omegaset,           /**< array to store the collected jobs */
   int*                  nelements,          /**< pointer to store the number of elements in omega set */
   int*                  est,                /**< pointer to store the earliest start time of the omega set */
   int*                  lct,                /**< pointer to store the latest start time of the omega set */
   int*                  energy              /**< pointer to store the energy of the omega set */
   )
{
   SCIP_BTNODE* left;
   SCIP_BTNODE* right;
   SCIP_NODEDATA* nodedata;
   SCIP_NODEDATA* leftdata;
   SCIP_NODEDATA* rightdata;

   assert(node != NULL);

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);

   /* check if the node is a leaf */
   if( SCIPbtnodeIsLeaf(node) )
   {
      assert(!nodedata->intheta);
      return;
   }

   left = SCIPbtnodeGetLeftchild(node);
   assert(left != NULL);

   leftdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(left);
   assert(leftdata != NULL);

   right = SCIPbtnodeGetRightchild(node);
   assert(right != NULL);

   rightdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(right);
   assert(rightdata != NULL);

   assert(nodedata->enveloplambda != -1);
   assert(rightdata->energytheta != -1);

   if( leftdata->enveloplambda >= 0 && nodedata->enveloplambda == leftdata->enveloplambda + rightdata->energytheta )
   {
      traceLambdaEnvelop(left, omegaset, nelements, est, lct, energy);
      collectThetaSubtree(right, omegaset, nelements, est, lct, energy);
   }
   else
   {
      if( leftdata->enveloptheta >= 0 && rightdata->energylambda >= 0
         && nodedata->enveloplambda == leftdata->enveloptheta + rightdata->energylambda )
      {
         traceThetaEnvelop(left, omegaset, nelements, est, lct, energy);
         traceLambdaEnergy(right, omegaset, nelements, est, lct, energy);
      }
      else
      {
         assert(rightdata->enveloplambda != -1);
         assert(nodedata->enveloplambda == rightdata->enveloplambda);
         traceLambdaEnvelop(right, omegaset, nelements, est, lct, energy);
      }
   }
}

/** compute the energy contribution by job which corresponds to the given leaf */
static
int computeEnergyContribution(
   SCIP_BTNODE*          node                /**< leaf */
   )
{
   SCIP_NODEDATA* nodedata;
   int duration;

   nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(node);
   assert(nodedata != NULL);
   assert(nodedata->var != NULL);

   duration = nodedata->duration - nodedata->leftadjust - nodedata->rightadjust;
   assert(duration > 0);

   SCIPdebugMessage("variable <%s>: loc=[%g,%g] glb=[%g,%g] (duration %d, demand %d)\n",
      SCIPvarGetName(nodedata->var), SCIPvarGetLbLocal(nodedata->var), SCIPvarGetUbLocal(nodedata->var),
      SCIPvarGetLbGlobal(nodedata->var), SCIPvarGetUbGlobal(nodedata->var), duration, nodedata->demand);

   /* return energy which is contributed by the start time variable */
   return nodedata->demand * duration;
}

/** comparison method for two node data w.r.t. the earliest start time */
static
SCIP_DECL_SORTPTRCOMP(compNodeEst)
{
   int est1;
   int est2;

   est1 = ((SCIP_NODEDATA*)SCIPbtnodeGetData((SCIP_BTNODE*)elem1))->est;
   est2 = ((SCIP_NODEDATA*)SCIPbtnodeGetData((SCIP_BTNODE*)elem2))->est;

   return (est1 - est2);
}

/** comparison method for two node data w.r.t. the latest completion time */
static
SCIP_DECL_SORTPTRCOMP(compNodedataLct)
{
   int lct1;
   int lct2;

   lct1 = ((SCIP_NODEDATA*)elem1)->lct;
   lct2 = ((SCIP_NODEDATA*)elem2)->lct;

   return (lct1 - lct2);
}


/** an overload was detected; initialized conflict analysis, add an initial reason
 *
 *  @note the conflict analysis is not performend, only the initialized SCIP_Bool pointer is set to TRUE
 */
static
SCIP_RETCODE analyzeConflictOverload(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_BTNODE**         leaves,             /**< responsible leaves for the overload */
   int                   capacity,           /**< cumulative capacity */
   int                   nleaves,            /**< number of responsible leaves */
   int                   est,                /**< earliest start time of the ...... */
   int                   lct,                /**< latest completly time of the .... */
   int                   reportedenergy,     /**< energy which already reported */
   SCIP_Bool             propest,            /**< should the earliest start times be propagated, otherwise the latest completion times */
   int                   shift,              /**< shift applied to all jobs before adding them to the tree */
   SCIP_Bool             usebdwidening,      /**< should bound widening be used during conflict analysis? */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation         /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   )
{
   int energy;
   int j;

   /* do nothing if conflict analysis is not applicable */
   if( !SCIPisConflictAnalysisApplicable(scip) )
      return SCIP_OKAY;

   SCIPdebugMessage("est=%d, lct=%d, propest %u, reportedenergy %d, shift %d\n", est, lct, propest, reportedenergy, shift);

   /* compute energy of initial time window */
   energy = (lct - est) * capacity;

   /* sort the start time variables which were added to search tree w.r.t. earliest start time */
   SCIPsortDownPtr((void**)leaves, compNodeEst, nleaves);

   /* collect the energy of the responsible leaves until the cumulative energy is large enough to detect an overload;
    * thereby, compute the time window of interest
    */
   for( j = 0; j < nleaves && reportedenergy <= energy; ++j )
   {
      SCIP_NODEDATA* nodedata;

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaves[j]);
      assert(nodedata != NULL);

      reportedenergy += computeEnergyContribution(leaves[j]);

      /* adjust energy if the earliest start time decrease */
      if( nodedata->est < est )
      {
         est = nodedata->est;
         energy = (lct - est) * capacity;
      }
   }
   assert(reportedenergy > energy);

   SCIPdebugMessage("time window [%d,%d) available energy %d, required energy %d\n", est, lct, energy, reportedenergy);

   /* initialize conflict analysis */
   SCIP_CALL( SCIPinitConflictAnalysis(scip) );

   /* flip earliest start time and latest completion time */
   if( !propest )
   {
      SCIPswapInts(&est, &lct);

      /* shift earliest start time and latest completion time */
      lct = shift - lct;
      est = shift - est;
   }
   else
   {
      /* shift earliest start time and latest completion time */
      lct = lct + shift;
      est = est + shift;
   }

   nleaves = j;

   /* report the variables and relax their bounds to final time interval [est,lct) which was been detected to be
    * overloaded
    */
   for( j = nleaves-1; j >= 0; --j )
   {
      SCIP_NODEDATA* nodedata;

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaves[j]);
      assert(nodedata != NULL);
      assert(nodedata->var != NULL);

      /* check if bound widening should be used */
      if( usebdwidening )
      {
         SCIP_CALL( SCIPaddConflictRelaxedUb(scip, nodedata->var, NULL, (SCIP_Real)(est - nodedata->leftadjust)) );
         SCIP_CALL( SCIPaddConflictRelaxedLb(scip, nodedata->var, NULL, (SCIP_Real)(lct - nodedata->duration + nodedata->rightadjust)) );
      }
      else
      {
         SCIP_CALL( SCIPaddConflictLb(scip, nodedata->var, NULL) );
         SCIP_CALL( SCIPaddConflictUb(scip, nodedata->var, NULL) );
      }

      if( explanation != NULL )
         explanation[nodedata->idx] = TRUE;
   }

   (*initialized) = TRUE;

   return SCIP_OKAY;
}

/** computes a new latest starting time of the job in 'respleaf' due to the energy consumption and stores the
 *  responsible interval bounds in *est_omega and *lct_omega
 */
static
int computeEstOmegaset(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   duration,           /**< duration of the job to move */
   int                   demand,             /**< demand of the job to move */
   int                   capacity,           /**< cumulative capacity */
   int                   est,                /**< earliest start time of the omega set */
   int                   lct,                /**< latest start time of the omega set */
   int                   energy              /**< energy of the omega set */
   )
{
   int newest;

   newest = 0;

   assert(scip != NULL);

   if( energy >  (capacity - demand) * (lct - est) )
   {
      if( energy + demand * duration > capacity * (lct - est) )
      {
         newest =  (int)SCIPfeasCeil(scip, (energy - (SCIP_Real)(capacity - demand) * (lct - est)) / (SCIP_Real)demand);
         newest += est;
      }
   }

   return newest;
}

/** propagates start time using an edge finding algorithm which is based on binary trees (theta lambda trees)
 *
 * @note The algorithm is based on the paper: Petr Vilim, "Edge Finding Filtering Algorithm for Discrete Cumulative
 *       Resources in O(kn log n)".  *I.P. Gent (Ed.): CP 2009, LNCS 5732, pp. 802-816, 2009.
 */
static
SCIP_RETCODE inferboundsEdgeFinding(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_CONS*            cons,               /**< constraint which is propagated */
   SCIP_BT*              tree,               /**< binary tree constaining the theta and lambda sets */
   SCIP_BTNODE**         leaves,             /**< array of all leaves for each job one */
   int                   capacity,           /**< cumulative capacity */
   int                   ncands,             /**< number of candidates */
   SCIP_Bool             propest,            /**< should the earliest start times be propagated, otherwise the latest completion times */
   int                   shift,              /**< shift applied to all jobs before adding them to the tree */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_NODEDATA* rootdata;
   int j;

   assert(!SCIPbtIsEmpty(tree));

   rootdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(SCIPbtGetRoot(tree));
   assert(rootdata != NULL);

   /* iterate over all added candidate (leaves) in non-increasing order w.r.t. their latest completion time */
   for( j = ncands-1; j >= 0 && !(*cutoff); --j )
   {
      SCIP_NODEDATA* nodedata;

      if( SCIPbtnodeIsRoot(leaves[j]) )
         break;

      nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaves[j]);
      assert(nodedata->est != -1);

      /* check if the root lambda envelop exeeds the available capacity */
      while( !(*cutoff) && rootdata->enveloplambda > capacity * nodedata->lct )
      {
         SCIP_BTNODE** omegaset;
         SCIP_BTNODE* leaf;
         SCIP_NODEDATA* leafdata;
         int nelements;
         int energy;
         int newest;
         int est;
         int lct;

         assert(!(*cutoff));

         /* find responsible leaf for the lambda envelope */
         leaf = findResponsibleLambdaLeafTraceEnvelop(SCIPbtGetRoot(tree));
         assert(leaf != NULL);
         assert(SCIPbtnodeIsLeaf(leaf));

         leafdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(leaf);
         assert(leafdata != NULL);
         assert(!leafdata->intheta);
         assert(leafdata->duration > 0);
         assert(leafdata->est >= 0);

         /* check if the job has to be removed since its latest completion is to large */
         if( leafdata->est + leafdata->duration >= nodedata->lct )
         {
            SCIP_CALL( deleteLambdaLeaf(scip, tree, leaf) );

            /* the root might changed therefore we need to collect the new root node data */
            rootdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(SCIPbtGetRoot(tree));
            assert(rootdata != NULL);

            continue;
         }

         /* compute omega set */
         SCIP_CALL( SCIPallocBufferArray(scip, &omegaset, ncands) );

         nelements = 0;
         est = INT_MAX;
         lct = INT_MIN;
         energy = 0;

         /* collect the omega set from theta set */
         traceLambdaEnvelop(SCIPbtGetRoot(tree), omegaset, &nelements, &est, &lct, &energy);
         assert(nelements > 0);
         assert(nelements < ncands);

         newest = computeEstOmegaset(scip, leafdata->duration, leafdata->demand, capacity, est, lct, energy);

         /* if the computed earliest start time is greater than the latest completion time of the omega set we detected an overload */
         if( newest > lct )
         {
            SCIPdebugMessage("an overload was detected duration edge-finder propagattion\n");

            /* analyze over load */
            SCIP_CALL( analyzeConflictOverload(scip, omegaset, capacity, nelements,  est, lct, 0, propest, shift,
                  conshdlrdata->usebdwidening, initialized, explanation) );
            (*cutoff) = TRUE;

            /* for the statistic we count the number of times a cutoff was detected due the edge-finder */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffedgefinder++ );
         }
         else if( newest > 0 )
         {
            SCIP_Bool infeasible;
            SCIP_Bool tightened;
            INFERINFO inferinfo;

            if( propest )
            {
               /* constuct inference information; store used propagation rule and the the time window of the omega set */
               inferinfo = getInferInfo(PROPRULE_2_EDGEFINDING, est + shift, lct + shift);

               SCIPdebugMessage("variable <%s> adjust lower bound from %g to %d\n",
                  SCIPvarGetName(leafdata->var), SCIPvarGetLbLocal(leafdata->var), newest + shift);

               SCIP_CALL( SCIPinferVarLbCons(scip, leafdata->var, (SCIP_Real)(newest + shift),
                     cons, inferInfoToInt(inferinfo), TRUE, &infeasible, &tightened) );

               /* for the statistic we count the number of times a lower bound was tightened due the edge-finder */
               SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nlbedgefinder++ );
            }
            else
            {
               /* constuct inference information; store used propagation rule and the the time window of the omega set */
               inferinfo = getInferInfo(PROPRULE_2_EDGEFINDING, shift - lct, shift - est);

               SCIPdebugMessage("variable <%s> adjust upper bound from %g to %d\n",
                  SCIPvarGetName(leafdata->var), SCIPvarGetUbLocal(leafdata->var), shift - newest - leafdata->duration);

               SCIP_CALL( SCIPinferVarUbCons(scip, leafdata->var, (SCIP_Real)(shift - newest - leafdata->duration),
                     cons, inferInfoToInt(inferinfo), TRUE, &infeasible, &tightened) );

               /* for the statistic we count the number of times a upper bound was tightened due the edge-finder */
               SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nubedgefinder++ );
            }

            /* adjust the earliest start time */
            if( tightened )
            {
               leafdata->est = newest;
               (*nchgbds)++;
            }

            if( infeasible )
            {
               /* initialize conflict analysis if conflict analysis is applicable */
               if( SCIPisConflictAnalysisApplicable(scip) )
               {
                  int i;

                  SCIPdebugMessage("edge-finder dectected an infeasibility\n");

                  SCIP_CALL( SCIPinitConflictAnalysis(scip) );

                  /* add lower and upper bound of variable which leads to the infeasibilty */
                  SCIP_CALL( SCIPaddConflictLb(scip, leafdata->var, NULL) );
                  SCIP_CALL( SCIPaddConflictUb(scip, leafdata->var, NULL) );

                  if( explanation != NULL )
                     explanation[leafdata->idx] = TRUE;

                  /* add lower and upper bound of variable which lead to the infeasibilty */
                  for( i = 0; i < nelements; ++i )
                  {
                     nodedata = (SCIP_NODEDATA*)SCIPbtnodeGetData(omegaset[i]);
                     assert(nodedata != NULL);

                     SCIP_CALL( SCIPaddConflictLb(scip, nodedata->var, NULL) );
                     SCIP_CALL( SCIPaddConflictUb(scip, nodedata->var, NULL) );

                     if( explanation != NULL )
                        explanation[nodedata->idx] = TRUE;
                  }

                  (*initialized) = TRUE;
               }

               (*cutoff) = TRUE;

               /* for the statistic we count the number of times a cutoff was detected due the edge-finder */
               SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffedgefinder++ );
            }
         }

         /* free omegaset array */
         SCIPfreeBufferArray(scip, &omegaset);

         /* delete responsible leaf from lambda */
         SCIP_CALL( deleteLambdaLeaf(scip, tree, leaf) );

         /* the root might changed therefore we need to collect the new root node data */
         rootdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(SCIPbtGetRoot(tree));
         assert(rootdata != NULL);
      }

      /* move current job j from the theta set into the lambda set */
      SCIP_CALL( moveNodeToLambda(scip, tree, leaves[j]) );
   }

   return SCIP_OKAY;
}

/** checks whether the instance is infeasible due to a overload within a certain time frame using the idea of theta trees
 *
 *  @note The algorithm is based on the paper: Petr Vilim, "Max Energy Filtering Algorithm for Discrete Cumulative
 *        Resources". In: Willem Jan van Hoeve and John N. Hooker (Eds.), Integration of AI and OR Techniques in
 *        Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2009), LNCS 5547, pp 294--308
 */
static
SCIP_RETCODE checkOverloadViaThetaTree(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated */
   SCIP_Bool             propest,            /**< should the earliest start times be propagated, otherwise the latest completion times */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_NODEDATA** nodedatas;
   SCIP_BTNODE** leaves;
   SCIP_BT* tree;

   int totalenergy;
   int nnodedatas;
   int ninsertcands;
   int ncands;

   int shift;
   int j;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(initialized != NULL);
   assert(cutoff != NULL);
   assert(*cutoff == FALSE);

   SCIPdebugMessage("check overload of cumulative condition of constraint <%s> (capacity %d)\n", SCIPconsGetName(cons), capacity);

   SCIP_CALL( SCIPallocBufferArray(scip, &nodedatas, 2*nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &leaves, nvars) );

   ncands = 0;
   totalenergy = 0;

   SCIP_CALL( SCIPbtCreate(&tree, SCIPblkmem(scip)) );

   /* compute the shift which we apply to compute .... latest completion time of all jobs */
   if( propest )
      shift = 0;
   else
   {
      shift = 0;

      /* compute the latest completion time of all jobs which define the shift we apply to run the algorithm for the
       * earliest start time propagation to handle the latest completion times
       */
      for( j = 0; j < nvars; ++j )
      {
         int lct;

         lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[j])) + durations[j];
         shift = MAX(shift, lct);
      }
   }

   /* collect earliest and latest completion times and ignore jobs which do not run completion within the effective
    * horizon
    */
   for( j = 0; j < nvars; ++j )
   {
      SCIP_NODEDATA* nodedata;
      SCIP_VAR* var;
      int duration;
      int leftadjust;
      int rightadjust;
      int energy;
      int est;
      int lct;

      var = vars[j];
      assert(var != NULL);

      duration = durations[j];
      assert(duration > 0);

      leftadjust = 0;
      rightadjust = 0;

      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration;

      /* adjust the duration, earliest start time, and latest completion time of jobs which do not lie completely in the
       * effective horizon [hmin,hmax)
       */
      if( conshdlrdata->useadjustedjobs )
      {
         if( est < hmin )
         {
            leftadjust = (hmin - est);
            est = hmin;
         }
         if( lct > hmax )
         {
            rightadjust = (lct - hmax);
            lct = hmax;
         }

         /* only consider jobs which have a (adjusted) duration greater than zero (the amound which will run defenetly
          * with the effective time horizon
          */
         if( duration - leftadjust - rightadjust <= 0 )
            continue;
      }
      else if( est < hmin || lct > hmax )
         continue;

      energy = demands[j] * (duration - leftadjust - rightadjust);
      assert(energy > 0);

      totalenergy += energy;

      /* flip earliest start time and latest completion time */
      if( !propest )
      {
         SCIPswapInts(&est, &lct);

         /* shift earliest start time and latest completion time */
         lct = shift - lct;
         est = shift - est;
      }
      else
      {
         /* shift earliest start time and latest completion time */
         lct = lct - shift;
         est = est - shift;
      }
      assert(est < lct);
      assert(est >= 0);
      assert(lct >= 0);

      /* create search node data */
      SCIP_CALL( createNodedata(scip, &nodedata) );

      /* initialize search node data */
      /* adjust earliest start time to make it unique in case several jobs have the same earliest start time */
      nodedata->key = est + j / (2.0 * nvars);
      nodedata->var = var;
      nodedata->est = est;
      nodedata->lct = lct;
      nodedata->demand = demands[j];
      nodedata->duration = duration;
      nodedata->leftadjust = leftadjust;
      nodedata->rightadjust = rightadjust;

      /* the envelop is the energy of the job plus the total amount of energy which is available in the time period
       * before that job can start, that is [0,est). The envelop is later used to compare the energy consumption of a
       * particular time interval [a,b] against the time interval [0,b].
       */
      nodedata->enveloptheta = capacity * est + energy;
      nodedata->energytheta = energy;
      nodedata->enveloplambda = -1;
      nodedata->energylambda = -1;

      nodedata->idx = j;
      nodedata->intheta = TRUE;

      nodedatas[ncands] = nodedata;
      ncands++;
   }

   nnodedatas = ncands;

   /* sort (non-decreasing) the jobs w.r.t. latest completion times */
   SCIPsortPtr((void**)nodedatas, compNodedataLct, ncands);

   ninsertcands = 0;

   /* iterate over all jobs in non-decreasing order of their latest completion times and add them to the theta set until
    * the root envelop detects an overload
    */
   for( j = 0; j < ncands; ++j )
   {
      SCIP_BTNODE* leaf;
      SCIP_NODEDATA* rootdata;

      /* check if the new job opens a time window which size is so large that it offers more energy than the total
       * energy of all candidate jobs. If so we skip that one.
       */
      if( (nodedatas[j]->lct - nodedatas[j]->est) * capacity >= totalenergy )
      {
         /* set the earliest start time to minus one to mark that candidate to be not used */
         nodedatas[j]->est = -1;
         continue;
      }

      /* create search node */
      SCIP_CALL( SCIPbtnodeCreate(tree, &leaf, (void*)nodedatas[j]) );

      /* insert new node into the theta set and updete the envelops */
      SCIP_CALL( insertThetanode(scip, tree, leaf, nodedatas, &nnodedatas) );
      assert(nnodedatas <= 2*nvars);

      /* move the inserted candidates together */
      leaves[ninsertcands] = leaf;
      ninsertcands++;

      assert(!SCIPbtIsEmpty(tree));
      rootdata = (SCIP_NODEDATA*)SCIPbtnodeGetData(SCIPbtGetRoot(tree));
      assert(rootdata != NULL);

      /* check if the theta set envelops exceeds the available capacity */
      if( rootdata->enveloptheta > capacity * nodedatas[j]->lct )
      {
         SCIPdebugMessage("detects cutoff due to overload in time window [?,%d) (ncands %d)\n", nodedatas[j]->lct, j);
         (*cutoff) = TRUE;

         /* for the statistic we count the number of times a cutoff was detected due the edge-finder */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutoffoverload++ );

         break;
      }
   }

   /* in case an overload was detected and the conflict analysis is applicable, create an initialize explanation */
   if( *cutoff )
   {
      int glbenery;
      int est;
      int lct;

      glbenery = 0;
      est = nodedatas[j]->est;
      lct = nodedatas[j]->lct;

      /* scan the remaining candidates for a global contributions within the time window of the last inserted candidate
       * which led to an overload
       */
      for( j = j+1; j < ncands; ++j )
      {
         SCIP_NODEDATA* nodedata;
         int duration;
         int glbest;
         int glblct;

         nodedata = nodedatas[j];
         assert(nodedata != NULL);

         duration = nodedata->duration - nodedata->leftadjust - nodedata->rightadjust;

         /* get latest start time */
         glbest = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(nodedata->var));
         glblct = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(nodedata->var)) + duration;

         /* check if parts of the jobs run with the time window defined by the last inserted job */
         if( glbest < est )
            duration -= (est - glbest);

         if( glblct > lct )
            duration -= (glblct - lct);

         if( duration > 0 )
         {
            glbenery += nodedata->demand * duration;

            if( explanation != NULL )
               explanation[nodedata->idx] = TRUE;
         }
      }

      /* analyze the overload */
      SCIP_CALL( analyzeConflictOverload(scip, leaves, capacity, ninsertcands, est, lct, glbenery, propest, shift,
            conshdlrdata->usebdwidening, initialized, explanation) );
   }
   else if( ninsertcands > 1 && conshdlrdata->efinfer )
   {
      /* if we have more than one job insterted and edge-finding should be performed we do it */
      SCIP_CALL( inferboundsEdgeFinding(scip, conshdlrdata, cons, tree, leaves, capacity, ninsertcands,
            propest, shift, initialized, explanation, nchgbds, cutoff) );
   }

   /* free the search nodes data */
   for( j = nnodedatas - 1; j >= 0; --j )
   {
      freeNodedata(scip, &nodedatas[j]);
   }

   /* free theta tree */
   SCIPbtFree(&tree);

   /* free buffer arrays */
   SCIPfreeBufferArray(scip, &leaves);
   SCIPfreeBufferArray(scip, &nodedatas);

   return SCIP_OKAY;
}

/** checks whether the instance is infeasible due to a overload within a certain time frame using the idea of theta trees
 *
 *  @note The algorithm is based on the paper: Petr Vilim, "Max Energy Filtering Algorithm for Discrete Cumulative
 *        Resources". In: Willem Jan van Hoeve and John N. Hooker (Eds.), Integration of AI and OR Techniques in
 *        Constraint Programming for Combinatorial Optimization Problems (CPAIOR 2009), LNCS 5547, pp 294--308
 */
static
SCIP_RETCODE propagateEdgeFinding(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   /* check if a cutoff was already detected */
   if( (*cutoff) )
      return SCIP_OKAY;

   /* check if at least the basic overload checking should be preformed */
   if( !conshdlrdata->efcheck )
      return SCIP_OKAY;

   /* check for overload, which may result in a cutoff */
   SCIP_CALL( checkOverloadViaThetaTree(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
         cons, TRUE, initialized, explanation, nchgbds, cutoff) );

   /* check if a cutoff was detected */
   if( (*cutoff) )
      return SCIP_OKAY;

   /* check if bound should be infer */
   if( !conshdlrdata->efinfer )
      return SCIP_OKAY;

   /* check for overload, which may result in a cutoff */
   SCIP_CALL( checkOverloadViaThetaTree(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
         cons, FALSE, initialized, explanation, nchgbds, cutoff) );

   return SCIP_OKAY;
}

/** checks if the constraint is redundant; that is the case if its capacity can never be exceeded; therefore we check
 *  with respect to the lower and upper bounds of the integer start time variables the maximum capacity usage for all
 *  event points
 */
static
SCIP_RETCODE consCheckRedundancy(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            redundant           /**< pointer to store whether this constraint is redundant */
   )
{

   SCIP_VAR* var;
   int* starttimes;              /* stores when each job is starting */
   int* endtimes;                /* stores when each job ends */
   int* startindices;            /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;              /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   int lb;
   int ub;
   int freecapacity;             /* remaining capacity */
   int curtime;                  /* point in time which we are just checking */
   int endindex;                 /* index of endsolvalues with: endsolvalues[endindex] > curtime */
   int njobs;
   int j;

   assert(scip != NULL);
   assert(redundant != NULL);

   (*redundant) = TRUE;

   /* if no activities are associated with this cumulative then this constraint is redundant */
   if( nvars == 0 )
      return SCIP_OKAY;

   assert(vars != NULL);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   njobs = 0;

   /* assign variables, start and endpoints to arrays */
   for( j = 0; j < nvars; ++j )
   {
      assert(durations[j] > 0);
      assert(demands[j] > 0);

      var = vars[j];
      assert(var != NULL);

      lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* check if jobs runs completely outside of the effective time horizon */
      if( lb >= hmax || ub + durations[j] <= hmin )
         continue;

      starttimes[njobs] = MAX(lb, hmin);
      startindices[njobs] = j;

      endtimes[njobs] =  MIN(ub + durations[j], hmax);
      endindices[njobs] = j;
      assert(starttimes[njobs] <= endtimes[njobs]);
      njobs++;
   }

   /* sort the arrays not-decreasing according to startsolvalues and endsolvalues (and sort the indices in the same way) */
   SCIPsortIntInt(starttimes, startindices, njobs);
   SCIPsortIntInt(endtimes, endindices, njobs);

   endindex = 0;
   freecapacity = capacity;

   /* check each start point of a job whether the capacity is violated or not */
   for( j = 0; j < njobs; ++j )
   {
      curtime = starttimes[j];

      /* stop checking, if time point is above hmax */
      if( curtime >= hmax )
         break;

      /* subtract all capacity needed up to this point */
      freecapacity -= demands[startindices[j]];
      while( j+1 < njobs && starttimes[j+1] == curtime )
      {
         ++j;
         freecapacity -= demands[startindices[j]];
      }

      /* free all capacity usages of jobs the are no longer running */
      while( endtimes[endindex] <= curtime )
      {
         freecapacity += demands[endindices[endindex]];
         ++endindex;
      }
      assert(freecapacity <= capacity);

      /* check freecapacity to be smaller than zero */
      if( freecapacity < 0 && curtime >= hmin )
      {
         (*redundant) = FALSE;
         break;
      }
   } /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   return SCIP_OKAY;
}

/** creates the worst case resource profile, that is, all jobs are inserted with the earliest start and latest
 *  completion time
 */
static
SCIP_RETCODE createCoreProfile(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PROFILE*         profile,            /**< resource profile */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   int v;

   /* insert all cores */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      SCIP_Bool infeasible;
      int duration;
      int demand;
      int begin;
      int end;
      int est;
      int lst;
      int pos;

      var = vars[v];
      assert(var != NULL);
      assert(SCIPisFeasIntegral(scip, SCIPvarGetLbLocal(var)));
      assert(SCIPisFeasIntegral(scip, SCIPvarGetUbLocal(var)));

      duration = durations[v];
      assert(duration > 0);

      demand =  demands[v];
      assert(demand > 0);

      /* collect earliest and latest start time */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* check if the job runs completely outside of the effective horizon [hmin, hmax); if so skip it */
      if( lst + duration <= hmin || est >= hmax )
         continue;

      /* compute core interval w.r.t. effective time horizon */
      begin = MAX(hmin, lst);
      end = MIN(hmax, est + duration);

      /* check if a core exists */
      if( begin >= end )
         continue;

      SCIPdebugMessage("variable <%s>[%d,%d] (duration %d, demand %d): add core [%d,%d)\n",
         SCIPvarGetName(var), est, lst, duration, demand, begin, end);

      /* insert the core into core resource profile (complexity O(log n)) */
      SCIP_CALL( SCIPprofileInsertCore(profile, begin, end, demand, &pos, &infeasible) );

      /* in case the insertion of the core leads to an infeasibility; start the conflict analysis */
      if( infeasible )
      {
         assert(begin <= SCIPprofileGetTime(profile, pos));
         assert(end > SCIPprofileGetTime(profile, pos));

         /* use conflict analysis to analysis the core insertion which was infeasible */
         SCIP_CALL( analyseInfeasibelCoreInsertion(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
               var, duration, demand, SCIPprofileGetTime(profile, pos), conshdlrdata->usebdwidening, initialized, explanation) );

         if( explanation != NULL )
            explanation[v] = TRUE;

         (*cutoff) = TRUE;

         /* for the statistic we count the number of times a cutoff was detected due the time-time */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ncutofftimetable++ );

         break;
      }
   }

   return SCIP_OKAY;
}

/** propagate the cumulative condition */
static
SCIP_RETCODE propagateCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PRESOLTIMING     presoltiming,       /**< current presolving timing */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which is propagated (needed to SCIPinferVar**Cons()) */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   SCIP_Bool*            redundant,          /**< pointer to store if the constraint is redundant */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_PROFILE* profile;

   assert(nchgbds != NULL);
   assert(initialized != NULL);
   assert(cutoff != NULL);
   assert(!(*cutoff));

   /**@todo avoid always sorting the variable array */

   /* check if the constraint is redundant */
   SCIP_CALL( consCheckRedundancy(scip, nvars, vars, durations, demands, capacity, hmin, hmax, redundant) );

   if( *redundant )
      return SCIP_OKAY;

   /* create an empty resource profile for profiling the cores of the jobs */
   SCIP_CALL( SCIPprofileCreate(&profile, capacity) );

   /* create core profile (compulsory parts) */
   SCIP_CALL( createCoreProfile(scip, conshdlrdata, profile, nvars, vars, durations, demands, capacity, hmin, hmax,
         initialized, explanation, cutoff) );

   /* propagate the job cores until nothing else can be detected */
   if( (presoltiming & SCIP_PRESOLTIMING_FAST) != 0 )
   {
      SCIP_CALL( propagateTimetable(scip, conshdlrdata, profile, nvars, vars, durations, demands, capacity, hmin, hmax, cons,
            nchgbds, initialized, explanation, cutoff) );
   }

   /* run edge finding propagator */
   if( (presoltiming & SCIP_PRESOLTIMING_EXHAUSTIVE) != 0 )
   {
      SCIP_CALL( propagateEdgeFinding(scip, conshdlrdata, nvars, vars, durations, demands, capacity, hmin, hmax,
            cons, initialized, explanation, nchgbds, cutoff) );
   }

   /* run time-table edge-finding propagator */
   if( (presoltiming & SCIP_PRESOLTIMING_MEDIUM) != 0 )
   {
      SCIP_CALL( propagateTTEF(scip, conshdlrdata, profile, nvars, vars, durations, demands, capacity, hmin, hmax, cons,
            nchgbds, initialized, explanation, cutoff) );
   }
   /* free resource profile */
   SCIPprofileFree(&profile);

   return SCIP_OKAY;
}

/** propagate the cumulative constraint */
static
SCIP_RETCODE propagateCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to propagate */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PRESOLTIMING     presoltiming,       /**< current presolving timing */
   int*                  nchgbds,            /**< pointer to store the number of bound changes */
   int*                  ndelconss,          /**< pointer to store the number of deleted constraints */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_Bool initialized;
   SCIP_Bool redundant;
   int oldnchgbds;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   oldnchgbds = *nchgbds;
   initialized = FALSE;
   redundant = FALSE;

   if( SCIPconsIsDeleted(cons) )
   {
      assert(SCIPinProbing(scip));
      return SCIP_OKAY;
   }

   /* if the constraint marked to be propagated, do nothing */
   if( consdata->propagated && SCIPgetStage(scip) != SCIP_STAGE_PRESOLVING )
      return SCIP_OKAY;

   SCIP_CALL( propagateCumulativeCondition(scip, conshdlrdata, presoltiming,
         consdata->nvars, consdata->vars, consdata->durations, consdata->demands, consdata->capacity,
         consdata->hmin, consdata->hmax, cons,
         nchgbds, &redundant, &initialized, NULL, cutoff) );

   if( redundant )
   {
      SCIPdebugMessage("%s deletes cumulative constraint <%s> since it is redundant\n",
         SCIPgetDepth(scip) == 0 ? "globally" : "locally", SCIPconsGetName(cons));

      if( !SCIPinProbing(scip) )
      {
         SCIP_CALL( SCIPdelConsLocal(scip, cons) );
         (*ndelconss)++;
      }
   }
   else
   {
      if( initialized )
      {
         /* run conflict analysis since it was initialized */
         assert(*cutoff == TRUE);
         SCIPdebugMessage("start conflict analysis\n");
         SCIP_CALL( SCIPanalyzeConflictCons(scip, cons, NULL) );
      }

      /* if successful, reset age of constraint */
      if( *cutoff || *nchgbds > oldnchgbds )
      {
         SCIP_CALL( SCIPresetConsAge(scip, cons) );
      }
      else
      {
         /* mark the constraint to be propagated */
         consdata->propagated = TRUE;
      }
   }

   return SCIP_OKAY;
}

/** it is dual feasible to remove the values {leftub+1, ..., rightlb-1} since SCIP current does not feature domain holes
 * we use the probing mode to check if one of the two branches is infeasible. If this is the case the dual redundant can
 * be realize as domain reduction. Otherwise we do nothing
 */
static
SCIP_RETCODE applyProbingVar(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR**            vars,               /**< problem variables */
   int                   nvars,              /**< number of problem variables */
   int                   probingpos,         /**< variable number to apply probing on */
   SCIP_Real             leftub,             /**< upper bound of probing variable in left branch */
   SCIP_Real             rightlb,            /**< lower bound of probing variable in right branch */
   SCIP_Real*            leftimpllbs,        /**< lower bounds after applying implications and cliques in left branch, or NULL */
   SCIP_Real*            leftimplubs,        /**< upper bounds after applying implications and cliques in left branch, or NULL */
   SCIP_Real*            leftproplbs,        /**< lower bounds after applying domain propagation in left branch */
   SCIP_Real*            leftpropubs,        /**< upper bounds after applying domain propagation in left branch */
   SCIP_Real*            rightimpllbs,       /**< lower bounds after applying implications and cliques in right branch, or NULL */
   SCIP_Real*            rightimplubs,       /**< upper bounds after applying implications and cliques in right branch, or NULL */
   SCIP_Real*            rightproplbs,       /**< lower bounds after applying domain propagation in right branch */
   SCIP_Real*            rightpropubs,       /**< upper bounds after applying domain propagation in right branch */
   int*                  nfixedvars,         /**< pointer to counter which is increased by the number of deduced variable fixations */
   SCIP_Bool*            success,            /**< buffer to store whether a probing succeed to dual fix the variable */
   SCIP_Bool*            cutoff              /**< buffer to store whether a cutoff is detected */
   )
{
   SCIP_VAR* var;
   SCIP_Bool tightened;

   assert(probingpos >= 0);
   assert(probingpos < nvars);
   assert(success != NULL);
   assert(cutoff != NULL);

   var = vars[probingpos];
   assert(var != NULL);
   assert(SCIPisGE(scip, leftub, SCIPvarGetLbLocal(var)));
   assert(SCIPisLE(scip, leftub, SCIPvarGetUbLocal(var)));
   assert(SCIPisGE(scip, rightlb, SCIPvarGetLbLocal(var)));
   assert(SCIPisLE(scip, rightlb, SCIPvarGetUbLocal(var)));

   (*success) = FALSE;

   if( SCIPinProbing(scip) || SCIPinRepropagation(scip) )
      return SCIP_OKAY;

   /* apply probing for the earliest start time (lower bound) of the variable (x <= est) */
   SCIP_CALL( SCIPapplyProbingVar(scip, vars, nvars, probingpos, SCIP_BOUNDTYPE_UPPER, leftub, -1,
         leftimpllbs, leftimplubs, leftproplbs, leftpropubs, cutoff) );

   if( (*cutoff) )
   {
      /* note that cutoff may occur if presolving has not been executed fully */
      SCIP_CALL( SCIPtightenVarLb(scip, var, rightlb, TRUE, cutoff, &tightened) );

      if( tightened )
      {
         (*success) =TRUE;
         (*nfixedvars)++;
      }

      return SCIP_OKAY;
   }

   /* note that probing can change the upper bound and thus the right branch may have been detected infeasible if
    * presolving has not been executed fully
    */
   if( SCIPisGT(scip, rightlb, SCIPvarGetUbLocal(var)) )
   {
      /* note that cutoff may occur if presolving has not been executed fully */
      SCIP_CALL( SCIPtightenVarUb(scip, var, leftub, TRUE, cutoff, &tightened) );

      if( tightened )
      {
         (*success) = TRUE;
         (*nfixedvars)++;
      }

      return SCIP_OKAY;
   }

   /* apply probing for the alternative lower bound of the variable (x <= alternativeubs[v]) */
   SCIP_CALL( SCIPapplyProbingVar(scip, vars, nvars, probingpos, SCIP_BOUNDTYPE_LOWER, rightlb, -1,
         rightimpllbs, rightimplubs, rightproplbs, rightpropubs, cutoff) );

   if( (*cutoff) )
   {
      /* note that cutoff may occur if presolving has not been executed fully */
      SCIP_CALL( SCIPtightenVarUb(scip, var, leftub, TRUE, cutoff, &tightened) );

      if( tightened )
      {
         (*success) =TRUE;
         (*nfixedvars)++;
      }

      return SCIP_OKAY;
   }

   return SCIP_OKAY;
}

/** is it possible, to round variable down w.r.t. objective function */
static
SCIP_RETCODE varMayRoundDown(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< problem variable */
   SCIP_Bool*            roundable           /**< pointer to store if the variable can be rounded down */
   )
{
   SCIP_Real objval;
   int scalar;

   assert(roundable != NULL);

   *roundable = TRUE;

   /* a fixed variable can be definition always be safely rounded */
   if( SCIPvarGetStatus(var) == SCIP_VARSTATUS_FIXED )
      return SCIP_OKAY;

   /* in case the variable is not active we need to check the object coefficient of the active variable */
   if( !SCIPvarIsActive(var) )
   {
      SCIP_VAR* actvar;
      int constant;

      actvar = var;

      SCIP_CALL( getActiveVar(scip, &actvar, &scalar, &constant) );
      assert(scalar != 0);

      objval = scalar * SCIPvarGetObj(actvar);
   }
   else
   {
      scalar = 1;
      objval = SCIPvarGetObj(var);
   }

   /* rounding the integer variable down is only a valid dual reduction if the object coefficient is zero or positive
    * (the transformed problem is always a minimization problem)
    *
    * @note that we need to check this condition w.r.t. active variable space
    */
   if( (scalar > 0 && SCIPisNegative(scip, objval)) || (scalar < 0 && SCIPisPositive(scip, objval)) )
      *roundable = FALSE;

   return SCIP_OKAY;
}

/** is it possible, to round variable up w.r.t. objective function */
static
SCIP_RETCODE varMayRoundUp(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< problem variable */
   SCIP_Bool*            roundable           /**< pointer to store if the variable can be rounded down */
   )
{
   SCIP_Real objval;
   int scalar;

   assert(roundable != NULL);

   *roundable = TRUE;

   /* a fixed variable can be definition always be safely rounded */
   if( SCIPvarGetStatus(var) == SCIP_VARSTATUS_FIXED )
      return SCIP_OKAY;

   /* in case the variable is not active we need to check the object coefficient of the active variable */
   if( !SCIPvarIsActive(var) )
   {
      SCIP_VAR* actvar;
      int constant;

      actvar = var;

      SCIP_CALL( getActiveVar(scip, &actvar, &scalar, &constant) );
      assert(scalar != 0);

      objval = scalar * SCIPvarGetObj(actvar);
   }
   else
   {
      scalar = 1;
      objval = SCIPvarGetObj(var);
   }

   /* rounding the integer variable up is only a valid dual reduction if the object coefficient is zero or negative
    * (the transformed problem is always a minimization problem)
    *
    * @note that we need to check this condition w.r.t. active variable space
    */
   if( (scalar > 0 && SCIPisPositive(scip, objval)) || (scalar < 0 && SCIPisNegative(scip, objval)) )
      *roundable = FALSE;

   return SCIP_OKAY;
}

/** For each variable we compute an alternative lower and upper bounds. That is, if the variable is not fixed to its
 *  lower or upper bound the next reasonable lower or upper bound would be this alternative bound (implying that certain
 *  values are not of interest). An alternative bound for a particular is only valied if the cumulative constarints are
 *  the only one locking this variable in the corresponding direction.
 */
static
SCIP_RETCODE computeAlternativeBounds(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           conss,              /**< array of cumulative constraint constraints */
   int                   nconss,             /**< number of cumulative constraints */
   SCIP_Bool             local,              /**< use local bounds effective horizon? */
   int*                  alternativelbs,     /**< alternative lower bounds */
   int*                  alternativeubs,     /**< alternative lower bounds */
   int*                  downlocks,          /**< number of constraints with down lock participating by the computation */
   int*                  uplocks             /**< number of constraints with up lock participating by the computation */
   )
{
   int nvars;
   int c;
   int v;

   for( c = 0; c < nconss; ++c )
   {
      SCIP_CONSDATA* consdata;
      SCIP_CONS* cons;
      SCIP_VAR* var;
      int hmin;
      int hmax;

      cons = conss[c];
      assert(cons != NULL);

      /* ignore constraints which are already deletet and those which are not check constraints */
      if( SCIPconsIsDeleted(cons) || !SCIPconsIsChecked(cons) )
         continue;

      consdata = SCIPconsGetData(cons);
      assert(consdata != NULL);
      assert(consdata->nvars > 1);

      /* compute the hmin and hmax */
      if( local )
      {
         SCIP_PROFILE* profile;

         /* create empty resource profile with infinity resource capacity */
         SCIP_CALL( SCIPprofileCreate(&profile, INT_MAX) );

         /* create worst case resource profile */
         SCIP_CALL( SCIPcreateWorstCaseProfile(scip, profile, consdata->nvars, consdata->vars, consdata->durations, consdata->demands) );

         hmin = SCIPcomputeHmin(scip, profile, consdata->capacity);
         hmax = SCIPcomputeHmax(scip, profile, consdata->capacity);

         /* free worst case profile */
         SCIPprofileFree(&profile);
      }
      else
      {
         hmin = consdata->hmin;
         hmax = consdata->hmax;
      }

      consdata = SCIPconsGetData(cons);
      assert(consdata != NULL);

      nvars = consdata->nvars;

      for( v = 0; v < nvars; ++v )
      {
         int scalar;
         int constant;
         int idx;

         var = consdata->vars[v];
         assert(var != NULL);

         /* multi-aggregated variables should appear here since we mark the variables to be not mutlt-aggregated */
         assert(SCIPvarGetStatus(var) != SCIP_VARSTATUS_MULTAGGR);

         /* ignore variable locally fixed variables */
         if( SCIPvarGetUbLocal(var) - SCIPvarGetLbLocal(var) < 0.5 )
            continue;


         SCIP_CALL( getActiveVar(scip, &var, &scalar, &constant) );
         idx = SCIPvarGetProbindex(var);
         assert(idx >= 0);

         /* first check lower bound fixing */
         if( consdata->downlocks[v] )
         {
            int ect;
            int est;

            /* the variable has a down locked */
            est = scalar * SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var)) + constant;
            ect = est + consdata->durations[v];

            if( ect <= hmin || hmin >= hmax )
               downlocks[idx]++;
            else if( est < hmin && alternativelbs[idx] >= (hmin + 1 - constant) / scalar )
            {
               alternativelbs[idx] = (hmin + 1 - constant) / scalar;
               downlocks[idx]++;
            }
         }

         /* second check upper bound fixing */
         if( consdata->uplocks[v] )
         {
            int duration;
            int lct;
            int lst;

            duration =  consdata->durations[v];

            /* the variable has a up lock locked */
            lst = scalar * SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + constant;
            lct = lst + duration;

            if( lst >= hmax || hmin >= hmax  )
               uplocks[idx]++;
            else if( lct > hmax && alternativeubs[idx] <= ((hmax - 1 - constant) / scalar) - duration )
            {
               alternativeubs[idx] = ((hmax - 1 - constant) / scalar)  - duration;
               uplocks[idx]++;
            }
         }
      }
   }

   return SCIP_OKAY;
}

/** apply all fixings which are given by the alternative bounds */
static
SCIP_RETCODE applyAlternativeBoundsFixing(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR**            vars,               /**< array of active variables */
   int                   nvars,              /**< number of active variables */
   int*                  alternativelbs,     /**< alternative lower bounds */
   int*                  alternativeubs,     /**< alternative lower bounds */
   int*                  downlocks,          /**< number of constraints with down lock participating by the computation */
   int*                  uplocks,            /**< number of constraints with up lock participating by the computation */
   int*                  nfixedvars,         /**< pointer to counter which is increased by the number of deduced variable fixations */
   SCIP_Bool*            cutoff              /**< buffer to store whether a cutoff is detected */
   )
{
   SCIP_Real* downimpllbs;
   SCIP_Real* downimplubs;
   SCIP_Real* downproplbs;
   SCIP_Real* downpropubs;
   SCIP_Real* upimpllbs;
   SCIP_Real* upimplubs;
   SCIP_Real* upproplbs;
   SCIP_Real* uppropubs;
   int v;

   /* get temporary memory for storing probing results */
   SCIP_CALL( SCIPallocBufferArray(scip, &downimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downpropubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &uppropubs, nvars) );

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      SCIP_Bool infeasible;
      SCIP_Bool fixed;
      SCIP_Bool roundable;
      int ub;
      int lb;

      var = vars[v];
      assert(var != NULL);

      lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* ignore fixed variables */
      if( ub - lb <= 0 )
         continue;


      if( SCIPvarGetNLocksDown(var) == downlocks[v] )
      {
         SCIP_CALL( varMayRoundDown(scip, var, &roundable) );

         if( roundable )
         {
            if( alternativelbs[v] > ub )
            {
               SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetLbLocal(var), &infeasible, &fixed) );
               assert(!infeasible);
               assert(fixed);

               (*nfixedvars)++;

               /* for the statistic we count the number of jobs which are dual fixed due the information of all cumulative
                * constraints
                */
               SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nallconsdualfixs++ );
            }
            else
            {
               SCIP_Bool success;

               /* In the current version SCIP, variable domains are single intervals. Meaning that domain holes or not
                * representable. To retrieve a potential dual reduction we using probing to check both branches. If one in
                * infeasible we can apply the dual reduction; otherwise we do nothing
                */
               SCIP_CALL( applyProbingVar(scip, vars, nvars, v, (SCIP_Real) lb, (SCIP_Real) alternativelbs[v],
                     downimpllbs, downimplubs, downproplbs, downpropubs, upimpllbs, upimplubs, upproplbs, uppropubs,
                     nfixedvars, &success, cutoff) );

               if( success )
               {
                  SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nallconsdualfixs++ );
               }

            }
         }
      }

      lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var));

      /* ignore fixed variables */
      if( ub - lb <= 0 )
         continue;

      if( SCIPvarGetNLocksUp(var) == uplocks[v] )
      {
         SCIP_CALL( varMayRoundUp(scip, var, &roundable) );

         if( roundable )
         {
            if( alternativeubs[v] < lb )
            {
               SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetUbLocal(var), &infeasible, &fixed) );
               assert(!infeasible);
               assert(fixed);

               (*nfixedvars)++;

               /* for the statistic we count the number of jobs which are dual fixed due the information of all cumulative
                * constraints
                */
               SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nallconsdualfixs++ );
            }
            else
            {
               SCIP_Bool success;

               /* In the current version SCIP, variable domains are single intervals. Meaning that domain holes or not
                * representable. To retrieve a potential dual reduction we using probing to check both branches. If one in
                * infeasible we can apply the dual reduction; otherwise we do nothing
                */
               SCIP_CALL( applyProbingVar(scip, vars, nvars, v, (SCIP_Real) alternativeubs[v], (SCIP_Real) ub,
                     downimpllbs, downimplubs, downproplbs, downpropubs, upimpllbs, upimplubs, upproplbs, uppropubs,
                     nfixedvars, &success, cutoff) );

               if( success )
               {
                  SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nallconsdualfixs++ );
               }
            }
         }
      }
   }

   /* free temporary memory */
   SCIPfreeBufferArray(scip, &uppropubs);
   SCIPfreeBufferArray(scip, &upproplbs);
   SCIPfreeBufferArray(scip, &upimplubs);
   SCIPfreeBufferArray(scip, &upimpllbs);
   SCIPfreeBufferArray(scip, &downpropubs);
   SCIPfreeBufferArray(scip, &downproplbs);
   SCIPfreeBufferArray(scip, &downimplubs);
   SCIPfreeBufferArray(scip, &downimpllbs);

   return SCIP_OKAY;
}

/** propagate all constraints together */
static
SCIP_RETCODE propagateAllConss(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_CONS**           conss,              /**< all cumulative constraint */
   int                   nconss,             /**< number of cumulative constraints */
   SCIP_Bool             local,              /**< use local bounds effective horizon? */
   int*                  nfixedvars,         /**< pointer to counter which is increased by the number of deduced variable fixations */
   SCIP_Bool*            cutoff,             /**< buffer to store whether a cutoff is detected */
   SCIP_Bool*            branched            /**< pointer to store if a branching was applied, or NULL to avoid branching */
   )
{  /*lint --e{715}*/
   SCIP_VAR** vars;
   int* downlocks;
   int* uplocks;
   int* alternativelbs;
   int* alternativeubs;
   int oldnfixedvars;
   int nvars;
   int v;

   if( SCIPinProbing(scip) || SCIPinRepropagation(scip) )
      return SCIP_OKAY;

   nvars = SCIPgetNVars(scip);
   oldnfixedvars = *nfixedvars;

   SCIP_CALL( SCIPduplicateBufferArray(scip, &vars, SCIPgetVars(scip), nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downlocks, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &uplocks, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &alternativelbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &alternativeubs, nvars) );

   /* initialize arrays */
   for( v = 0; v < nvars; ++v )
   {
      downlocks[v] = 0;
      uplocks[v] = 0;
      alternativelbs[v] = INT_MAX;
      alternativeubs[v] = INT_MIN;
   }

   /* compute alternative bounds */
   SCIP_CALL( computeAlternativeBounds(scip, conss, nconss, local, alternativelbs, alternativeubs, downlocks, uplocks) );

   /* apply fixing which result of the alternative bounds directly */
   SCIP_CALL( applyAlternativeBoundsFixing(scip, vars, nvars, alternativelbs, alternativeubs, downlocks, uplocks,
         nfixedvars, cutoff) );

   if( !(*cutoff) && oldnfixedvars == *nfixedvars && branched != NULL )
   {
      SCIP_CALL( applyAlternativeBoundsBranching(scip, vars, nvars, alternativelbs, alternativeubs, downlocks, uplocks, branched) );
   }

   /* free all buffers */
   SCIPfreeBufferArray(scip, &alternativeubs);
   SCIPfreeBufferArray(scip, &alternativelbs);
   SCIPfreeBufferArray(scip, &uplocks);
   SCIPfreeBufferArray(scip, &downlocks);
   SCIPfreeBufferArray(scip, &vars);

   return SCIP_OKAY;
}

/**@} */

/**@name Linear relaxations
 *
 * @{
 */

/** creates covering cuts for jobs violating resource constraints */
static
SCIP_RETCODE createCoverCutsTimepoint(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   int*                  startvalues,        /**< upper bounds on finishing time per job for activities from 0,..., nactivities -1 */
   int                   time                /**< at this point in time covering constraints are valid */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_ROW* row;
   int* flexibleids;
   int* demands;

   char rowname[SCIP_MAXSTRLEN];

   int remainingcap;
   int smallcoversize;    /* size of a small cover */
   int bigcoversize;    /* size of a big cover */
   int nvars;

   int nflexible;
   int sumdemand;            /* demand of all jobs up to a certain index */
   int j;

   assert(cons != NULL);

   /* get constraint data structure */
   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL );

   nvars = consdata->nvars;

   /* sort jobs according to demands */
   SCIP_CALL( SCIPallocBufferArray(scip, &demands, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &flexibleids, nvars) );

   nflexible = 0;
   remainingcap = consdata->capacity;

   /* get all jobs intersecting point 'time' with their bounds */
   for( j = 0; j < nvars; ++j )
   {
      int ub;

      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[j]));

      /* only add jobs to array if they intersect with point 'time' */
      if( startvalues[j] <= time && ub + consdata->durations[j] > time )
      {
         /* if job is fixed, capacity has to be decreased */
         if( startvalues[j] == ub )
         {
            remainingcap -= consdata->demands[j];
         }
         else
         {
            demands[nflexible] = consdata->demands[j];
            flexibleids[nflexible] = j;
            ++nflexible;
         }
      }
   }
   assert(remainingcap >= 0);

   /* sort demands and job ids */
   SCIPsortIntInt(demands, flexibleids, nflexible);

   /*
    * version 1:
    * D_j := sum_i=0,...,j  d_i, finde j maximal, so dass D_j <= remainingcap
    * erzeuge cover constraint
    *
    */

   /* find maximum number of jobs that can run in parallel (-->coversize = j) */
   sumdemand = 0;
   j = 0;

   while( j < nflexible && sumdemand <= remainingcap )
   {
      sumdemand += demands[j];
      j++;
   }

   /* j jobs form a conflict, set coversize to 'j - 1' */
   bigcoversize = j-1;
   assert(sumdemand > remainingcap);
   assert(bigcoversize < nflexible);

   /* - create a row for all jobs and their binary variables.
    * - at most coversize many binary variables of jobs can be set to one
    */

   /* construct row name */
   (void)SCIPsnprintf(rowname, SCIP_MAXSTRLEN, "capacity_coverbig_%d", time);
   SCIP_CALL( SCIPcreateEmptyRowCons(scip, &row, SCIPconsGetHdlr(cons), rowname, -SCIPinfinity(scip), (SCIP_Real)bigcoversize,
         SCIPconsIsLocal(cons), SCIPconsIsModifiable(cons), TRUE) );
   SCIP_CALL( SCIPcacheRowExtensions(scip, row) );

   for( j = 0; j < nflexible; ++j )
   {
      SCIP_VAR** binvars;
      int* vals;
      int nbinvars;
      int idx;
      int start;
      int end;
      int lb;
      int ub;
      int b;

      idx = flexibleids[j];

      /* get and add binvars into var array */
      SCIP_CALL( SCIPgetBinvarsLinking(scip, consdata->linkingconss[idx], &binvars, &nbinvars) );
      assert(nbinvars != 0);

      vals = SCIPgetValsLinking(scip, consdata->linkingconss[idx]);
      assert(vals != NULL);

      lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[idx]));
      ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[idx]));

      /* compute start and finishing time */
      start = time - consdata->durations[idx] + 1;
      end =  MIN(time, ub);

      /* add all neccessary binary variables */
      for( b = 0; b < nbinvars; ++b )
      {
         if( vals[b] < start || vals[b] < lb )
            continue;

         if( vals[b] > end )
            break;

         assert(binvars[b] != NULL);
         SCIP_CALL( SCIPaddVarToRow(scip, row, binvars[b], 1.0) );
      }
   }

   /* insert and release row */
   SCIP_CALL( SCIPflushRowExtensions(scip, row) );

   if( consdata->bcoverrowssize == 0 )
   {
      consdata->bcoverrowssize = 10;
      SCIP_CALL( SCIPallocBlockMemoryArray(scip, &consdata->bcoverrows, consdata->bcoverrowssize) );
   }
   if( consdata->nbcoverrows == consdata->bcoverrowssize )
   {
      consdata->bcoverrowssize *= 2;
      SCIP_CALL( SCIPreallocBlockMemoryArray(scip, &consdata->bcoverrows, consdata->nbcoverrows, consdata->bcoverrowssize) );
   }

   consdata->bcoverrows[consdata->nbcoverrows] = row;
   consdata->nbcoverrows++;

   /*
    * version 2:
    * D_j := sum_i=j,...,0  d_i, finde j minimal, so dass D_j <= remainingcap
    * erzeuge cover constraint und fuege alle jobs i hinzu, mit d_i = d_largest
    */
   /* find maximum number of jobs that can run in parallel (= coversize -1) */
   sumdemand = 0;
   j = nflexible -1;
   while( sumdemand <= remainingcap )
   {
      assert(j >= 0);
      sumdemand += demands[j];
      j--;
   }

   smallcoversize = nflexible - (j + 1) - 1;
   while( j > 0 && demands[j] == demands[nflexible-1] )
      --j;

   assert(smallcoversize < nflexible);

   if( smallcoversize != 1 || smallcoversize != nflexible - (j + 1) - 1 )
   {
      /* construct row name */
      (void)SCIPsnprintf(rowname, SCIP_MAXSTRLEN, "capacity_coversmall_%d", time);
      SCIP_CALL( SCIPcreateEmptyRowCons(scip, &row, SCIPconsGetHdlr(cons), rowname, -SCIPinfinity(scip), (SCIP_Real)smallcoversize,
            SCIPconsIsLocal(cons), SCIPconsIsModifiable(cons), TRUE) );
      SCIP_CALL( SCIPcacheRowExtensions(scip, row) );

      /* filter binary variables for each unfixed job */
      for( j = j + 1; j < nflexible; ++j )
      {
         SCIP_VAR** binvars;
         int* vals;
         int nbinvars;
         int idx;
         int start;
         int end;
         int lb;
         int ub;
         int b;

         idx = flexibleids[j];

         /* get and add binvars into var array */
         SCIP_CALL( SCIPgetBinvarsLinking(scip, consdata->linkingconss[idx], &binvars, &nbinvars) );
         assert(nbinvars != 0);

         vals = SCIPgetValsLinking(scip, consdata->linkingconss[idx]);
         assert(vals != NULL);

         lb = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[idx]));
         ub = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[idx]));

         /* compute start and finishing time */
         start = time - consdata->durations[idx] + 1;
         end =  MIN(time, ub);

         /* add all neccessary binary variables */
         for( b = 0; b < nbinvars; ++b )
         {
            if( vals[b] < start || vals[b] < lb )
               continue;

            if( vals[b] > end )
               break;

            assert(binvars[b] != NULL);
            SCIP_CALL( SCIPaddVarToRow(scip, row, binvars[b], 1.0) );
         }
      }

      /* insert and release row */
      SCIP_CALL( SCIPflushRowExtensions(scip, row) );
      if( consdata->scoverrowssize == 0 )
      {
         consdata->scoverrowssize = 10;
         SCIP_CALL( SCIPallocBlockMemoryArray(scip, &consdata->scoverrows, consdata->scoverrowssize) );
      }
      if( consdata->nscoverrows == consdata->scoverrowssize )
      {
         consdata->scoverrowssize *= 2;
         SCIP_CALL( SCIPreallocBlockMemoryArray(scip, &consdata->scoverrows, consdata->nscoverrows, consdata->scoverrowssize) );
      }

      consdata->scoverrows[consdata->nscoverrows] = row;
      consdata->nscoverrows++;
   }

   /* free buffer arrays */
   SCIPfreeBufferArray(scip, &flexibleids);
   SCIPfreeBufferArray(scip, &demands);

   return SCIP_OKAY;
}

/** method to construct cover cuts for all points in time */
static
SCIP_RETCODE createCoverCuts(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint to be separated */
   )
{
   SCIP_CONSDATA* consdata;

   int* startvalues;        /* stores when each job is starting */
   int* endvalues;          /* stores when each job ends */
   int* startvaluessorted;  /* stores when each job is starting */
   int* endvaluessorted;    /* stores when each job ends */
   int* startindices;     /* we sort the startvalues, so we need to know wich index of a job it corresponds to */
   int* endindices;       /* we sort the endvalues, so we need to know wich index of a job it corresponds to */

   int nvars;               /* number of jobs for this constraint */
   int freecapacity;        /* remaining capacity */
   int curtime;             /* point in time which we are just checking */
   int endidx;              /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int hmin;
   int hmax;

   int j;
   int t;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   /* if no activities are associated with this resource then this constraint is redundant */
   if( consdata->vars == NULL )
      return SCIP_OKAY;

   nvars = consdata->nvars;
   hmin = consdata->hmin;
   hmax = consdata->hmax;

   SCIP_CALL( SCIPallocBufferArray(scip, &startvalues, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endvalues, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startvaluessorted, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endvaluessorted, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   /* assign start and endpoints to arrays */
   for ( j = 0; j < nvars; ++j )
   {
      startvalues[j] = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[j]));
      startvaluessorted[j] = startvalues[j];

      endvalues[j] = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[j])) + consdata->durations[j];
      endvaluessorted[j] = endvalues[j];

      startindices[j] = j;
      endindices[j] = j;
   }

   /* sort the arrays not-decreasing according to startsolvalues and endsolvalues
    * (and sort the indices in the same way) */
   SCIPsortIntInt(startvaluessorted, startindices, nvars);
   SCIPsortIntInt(endvaluessorted, endindices, nvars);

   endidx = 0;
   freecapacity = consdata->capacity;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      curtime = startvaluessorted[j];
      if( curtime >= hmax )
         break;

      /* subtract all capacity needed up to this point */
      freecapacity -= consdata->demands[startindices[j]];

      while( j+1 < nvars && startvaluessorted[j+1] == curtime )
      {
         ++j;
         freecapacity -= consdata->demands[startindices[j]];
      }

      /* free all capacity usages of jobs the are no longer running */
      while( endidx < nvars && curtime >= endvaluessorted[endidx] )
      {
         freecapacity += consdata->demands[endindices[endidx]];
         ++endidx;
      }

      assert(freecapacity <= consdata->capacity);
      assert(endidx <= nvars);

      /* --> endindex - points to the next job which will finish
       *     j        - points to the last job that has been released
       */


      /* check freecapacity to be smaller than zero
       * then we will add cover constraints to the MIP
       */
      if( freecapacity < 0 && curtime >= hmin )
      {
         int nextprofilechange;

         /* we can create covering constraints for each pint in time in interval [curtime; nextprofilechange[ */
         if( j < nvars-1 )
            nextprofilechange = MIN( startvaluessorted[j+1], endvaluessorted[endidx] );
         else
            nextprofilechange = endvaluessorted[endidx];

         nextprofilechange = MIN(nextprofilechange, hmax);

         for( t = curtime; t < nextprofilechange; ++t )
         {
            SCIPdebugMessage("add cover constraint for time %d\n", curtime);

            /* create covering constraint */
            SCIP_CALL( createCoverCutsTimepoint(scip, cons, startvalues, t)  );

         }
      } /* end if freecapacity > 0 */

   } /*lint --e{850}*/

   consdata->covercuts = TRUE;

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endvaluessorted);
   SCIPfreeBufferArray(scip, &startvaluessorted);
   SCIPfreeBufferArray(scip, &endvalues);
   SCIPfreeBufferArray(scip, &startvalues);

   return SCIP_OKAY;
}

/** this method creates a row for time point curtime which insures the capacity restriction of the cumulative
 *  constraint
 */
static
SCIP_RETCODE createCapacityRestriction(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished,          /**< number of jobs that finished before curtime or at curtime */
   SCIP_Bool             cutsasconss         /**< should the cumulative constraint create the cuts as constraints? */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR** binvars;
   int* coefs;
   int nbinvars;
   char name[SCIP_MAXSTRLEN];
   int capacity;
   int b;

   assert(nstarted > nfinished);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(consdata->nvars > 0);

   capacity = consdata->capacity;
   assert(capacity > 0);

   nbinvars = 0;
   SCIP_CALL( collectBinaryVars(scip, consdata, &binvars, &coefs, &nbinvars, startindices, curtime, nstarted, nfinished) );

   /* construct row name */
   (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "%s_%d[%d]", SCIPconsGetName(cons), nstarted-1, curtime);

   if( cutsasconss )
   {
      SCIP_CONS* lincons;

      /* create knapsack constraint for the given time point */
      SCIP_CALL( SCIPcreateConsKnapsack(scip, &lincons, name, 0, NULL, NULL, (SCIP_Longint)(capacity),
            TRUE, TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, FALSE, TRUE, FALSE) );

      for( b = 0; b < nbinvars; ++b )
      {
         SCIP_CALL( SCIPaddCoefKnapsack(scip, lincons, binvars[b], (SCIP_Longint)coefs[b]) );
      }

      /* add and release the new constraint */
      SCIP_CALL( SCIPaddCons(scip, lincons) );
      SCIP_CALL( SCIPreleaseCons(scip, &lincons) );
   }
   else
   {
      SCIP_ROW* row;

      SCIP_CALL( SCIPcreateEmptyRowCons(scip, &row, SCIPconsGetHdlr(cons), name, -SCIPinfinity(scip), (SCIP_Real)capacity, FALSE, FALSE, SCIPconsIsRemovable(cons)) );
      SCIP_CALL( SCIPcacheRowExtensions(scip, row) );

      for( b = 0; b < nbinvars; ++b )
      {
         SCIP_CALL( SCIPaddVarToRow(scip, row, binvars[b], (SCIP_Real)coefs[b]) );
      }

      SCIP_CALL( SCIPflushRowExtensions(scip, row) );
      SCIPdebug( SCIP_CALL(SCIPprintRow(scip, row, NULL)) );

      if( consdata->demandrowssize == 0 )
      {
         consdata->demandrowssize = 10;
         SCIP_CALL( SCIPallocBlockMemoryArray(scip, &consdata->demandrows, consdata->demandrowssize) );
      }
      if( consdata->ndemandrows == consdata->demandrowssize )
      {
         consdata->demandrowssize *= 2;
         SCIP_CALL( SCIPreallocBlockMemoryArray(scip, &consdata->demandrows, consdata->ndemandrows, consdata->demandrowssize) );
      }

      consdata->demandrows[consdata->ndemandrows] = row;
      consdata->ndemandrows++;
   }

   SCIPfreeBufferArrayNull(scip, &binvars);
   SCIPfreeBufferArrayNull(scip, &coefs);

   return SCIP_OKAY;
}

/** this method checks how many cumulatives can run at most at one time if this is greater than the capacity it creates
 *  row
 */
static
SCIP_RETCODE consCapacityConstraintsFinder(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   SCIP_Bool             cutsasconss         /**< should the cumulative constraint create the cuts as constraints? */
   )
{
   SCIP_CONSDATA* consdata;

   int* starttimes;         /* stores when each job is starting */
   int* endtimes;           /* stores when each job ends */
   int* startindices;       /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;         /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   int nvars;               /* number of activities for this constraint */
   int freecapacity;        /* remaining capacity */
   int curtime;             /* point in time which we are just checking */
   int endindex;            /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int hmin;
   int hmax;

   int j;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   /* if no activities are associated with this cumulative then this constraint is redundant */
   if( nvars == 0 )
      return SCIP_OKAY;

   assert(consdata->vars != NULL);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   SCIPdebugMessage("create sorted event points for cumulative constraint <%s> with %d jobs\n",
      SCIPconsGetName(cons), nvars);

   /* create event point arrays */
   createSortedEventpoints(scip, nvars, consdata->vars, consdata->durations,
      starttimes, endtimes, startindices, endindices, FALSE);

   endindex = 0;
   freecapacity = consdata->capacity;
   hmin = consdata->hmin;
   hmax = consdata->hmax;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      curtime = starttimes[j];
      SCIPdebugMessage("look at %d-th job with start %d\n", j, curtime);

      if( curtime >= hmax )
         break;

      /* remove the capacity requirments for all job which start at the curtime */
      subtractStartingJobDemands(consdata, curtime, starttimes, startindices, &freecapacity, &j, nvars);

      /* add the capacity requirments for all job which end at the curtime */
      addEndingJobDemands(consdata, curtime, endtimes, endindices, &freecapacity, &endindex, nvars);

      assert(freecapacity <= consdata->capacity);
      assert(endindex <= nvars);

      /* endindex - points to the next job which will finish */
      /* j - points to the last job that has been released */

      /* if free capacity is smaller than zero, then add rows to the LP */
      if( freecapacity < 0 && curtime >= hmin )
      {
         int nextstarttime;
         int t;

         /* step forward until next job is released and see whether capacity constraint is met or not */
         if( j < nvars-1 )
            nextstarttime = starttimes[j+1];
         else
            nextstarttime = endtimes[nvars-1];

         nextstarttime = MIN(nextstarttime, hmax);

         /* create capacity restriction row for current event point */
         SCIP_CALL( createCapacityRestriction(scip, cons, startindices, curtime, j+1, endindex, cutsasconss) );

         /* create for all points in time between the current event point and next start event point a row if the free
          * capacity is still smaller than zero  */
         for( t = curtime+1 ; t < nextstarttime; ++t )
         {
            /* add the capacity requirments for all job which end at the curtime */
            addEndingJobDemands(consdata, t, endtimes, endindices, &freecapacity, &endindex, nvars);

            if( freecapacity < 0 )
            {
               /* add constraint */
               SCIPdebugMessage("add capacity constraint at time %d\n", t);

               /* create capacity restriction row */
               SCIP_CALL( createCapacityRestriction(scip, cons, startindices, t, j+1, endindex, cutsasconss) );
            }
            else
               break;
         }
      }
   } /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   return SCIP_OKAY;
}

/** creates LP rows corresponding to cumulative constraint; therefore, check each point in time if the maximal needed
 *  capacity is larger than the capacity of the cumulative constraint
 *  - for each necessary point in time:
 *
 *    sum_j sum_t demand_j * x_{j,t} <= capacity
 *
 *    where x(j,t) is the binary variables of job j at time t
 */
static
SCIP_RETCODE createRelaxation(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   SCIP_Bool             cutsasconss         /**< should the cumulative constraint create the cuts as constraints? */
   )
{
   SCIP_CONSDATA* consdata;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(consdata->demandrows == NULL);
   assert(consdata->ndemandrows == 0);

   /* collect the linking constraints */
   if( consdata->linkingconss == NULL )
   {
      SCIP_CALL( consdataCollectLinkingCons(scip, consdata) );
   }

   SCIP_CALL( consCapacityConstraintsFinder(scip, cons, cutsasconss) );

   /* switch of separation for the cumulative constraint if linear constraints are add as cuts */
   if( cutsasconss )
   {
      if( SCIPconsIsInitial(cons) )
      {
         SCIP_CALL( SCIPsetConsInitial(scip, cons, FALSE) );
      }
      if( SCIPconsIsSeparated(cons) )
      {
         SCIP_CALL( SCIPsetConsSeparated(scip, cons, FALSE) );
      }
      if( SCIPconsIsEnforced(cons) )
      {
         SCIP_CALL( SCIPsetConsEnforced(scip, cons, FALSE) );
      }
   }

   return SCIP_OKAY;
}

/** adds linear relaxation of cumulative constraint to the LP */
static
SCIP_RETCODE addRelaxation(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   SCIP_Bool             cutsasconss         /**< should the cumulative constraint create the cuts as constraints? */
   )
{
   SCIP_CONSDATA* consdata;
   int r;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   if( consdata->demandrows == NULL )
   {
      SCIP_CALL( createRelaxation(scip, cons, cutsasconss) );
   }

   for( r = 0; r < consdata->ndemandrows; ++r )
   {
      if( !SCIProwIsInLP(consdata->demandrows[r]) )
      {
         SCIP_Bool infeasible;

         assert(consdata->demandrows[r] != NULL);
         SCIP_CALL( SCIPaddCut(scip, NULL, consdata->demandrows[r], FALSE, &infeasible) );
         assert( ! infeasible );  /* this function is only called by initlp -> the cut should be feasible */
      }
   }

   return SCIP_OKAY;
}

/** checks constraint for violation, and adds it as a cut if possible */
static
SCIP_RETCODE separateConsBinaryRepresentation(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint to be separated */
   SCIP_SOL*             sol,                /**< primal CIP solution, NULL for current LP solution */
   SCIP_Bool*            separated,          /**< pointer to store TRUE, if a cut was found */
   SCIP_Bool*            cutoff              /**< whether a cutoff has been detected */
   )
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;
   int ncuts;
   int r;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(separated != NULL);
   assert(cutoff != NULL);

   *separated = FALSE;
   *cutoff = FALSE;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   SCIPdebugMessage("separate cumulative constraint <%s>\n", SCIPconsGetName(cons));

   if( consdata->demandrows == NULL )
   {
      SCIP_CALL( createRelaxation(scip, cons, FALSE) );
   }

   ncuts = 0;

   /* check each row that is not contained in LP */
   for( r = 0; r < consdata->ndemandrows; ++r )
   {
      if( !SCIProwIsInLP(consdata->demandrows[r]) )
      {
         SCIP_Real feasibility;

         if( sol != NULL )
            feasibility = SCIPgetRowSolFeasibility(scip, consdata->demandrows[r], sol);
         else
            feasibility = SCIPgetRowLPFeasibility(scip, consdata->demandrows[r]);

         if( SCIPisFeasNegative(scip, feasibility) )
         {
            SCIP_CALL( SCIPaddCut(scip, sol,  consdata->demandrows[r], FALSE, cutoff) );
            if ( *cutoff )
            {
               SCIP_CALL( SCIPresetConsAge(scip, cons) );
               return SCIP_OKAY;
            }
            *separated = TRUE;
            ncuts++;
         }
      }
   }

   if( ncuts > 0 )
   {
      SCIPdebugMessage("cumulative constraint <%s> separated %d cuts\n",  SCIPconsGetName(cons), ncuts);

      /* if successful, reset age of constraint */
      SCIP_CALL( SCIPresetConsAge(scip, cons) );
      (*separated) = TRUE;
   }

   return SCIP_OKAY;
}

/** checks constraint for violation, and adds it as a cut if possible */
static
SCIP_RETCODE separateCoverCutsCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< logic or constraint to be separated */
   SCIP_SOL*             sol,                /**< primal CIP solution, NULL for current LP solution */
   SCIP_Bool*            separated,          /**< pointer to store TRUE, if a cut was found */
   SCIP_Bool*            cutoff              /**< whether a cutoff has been detected */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_ROW* row;
   SCIP_Real minfeasibility;
   int r;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(separated != NULL);
   assert(cutoff != NULL);

   *separated = FALSE;
   *cutoff = FALSE;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   SCIPdebugMessage("separate cumulative constraint <%s>\n", SCIPconsGetName(cons));

   /* collect the linking constraints */
   if( consdata->linkingconss == NULL )
   {
      SCIP_CALL( consdataCollectLinkingCons(scip, consdata) );
   }

   if( !consdata->covercuts )
   {
      SCIP_CALL( createCoverCuts(scip, cons) );
   }

   row = NULL;
   minfeasibility = SCIPinfinity(scip);

   /* check each row of small covers that is not contained in LP */
   for( r = 0; r < consdata->nscoverrows; ++r )
   {
      if( !SCIProwIsInLP(consdata->scoverrows[r]) )
      {
         SCIP_Real feasibility;

         assert(consdata->scoverrows[r] != NULL);
         if( sol != NULL )
            feasibility = SCIPgetRowSolFeasibility(scip, consdata->scoverrows[r], sol);
         else
            feasibility = SCIPgetRowLPFeasibility(scip, consdata->scoverrows[r]);

         if( minfeasibility > feasibility )
         {
            minfeasibility = feasibility;
            row =  consdata->scoverrows[r];
         }
      }
   }

   if( SCIPisFeasNegative(scip, minfeasibility) )
   {
      SCIPdebugMessage("cumulative constraint <%s> separated 1 cover cut with feasibility %g\n",
         SCIPconsGetName(cons), minfeasibility);

      assert(row != NULL);
      SCIP_CALL( SCIPaddCut(scip, sol, row, FALSE, cutoff) );
      SCIP_CALL( SCIPresetConsAge(scip, cons) );
      if ( *cutoff )
         return SCIP_OKAY;
      (*separated) = TRUE;
   }

   minfeasibility = SCIPinfinity(scip);
   row = NULL;

   /* check each row of small covers that is not contained in LP */
   for( r = 0; r < consdata->nbcoverrows; ++r )
   {
      if( !SCIProwIsInLP(consdata->bcoverrows[r]) )
      {
         SCIP_Real feasibility;

         assert(consdata->bcoverrows[r] != NULL);
         if( sol != NULL )
            feasibility = SCIPgetRowSolFeasibility(scip, consdata->bcoverrows[r], sol);
         else
            feasibility = SCIPgetRowLPFeasibility(scip, consdata->bcoverrows[r]);

         if( minfeasibility > feasibility )
         {
            minfeasibility = feasibility;
            row =  consdata->bcoverrows[r];
         }
      }
   }

   if( SCIPisFeasNegative(scip, minfeasibility) )
   {
      SCIPdebugMessage("cumulative constraint <%s> separated 1 cover cut with feasibility %g\n",
         SCIPconsGetName(cons), minfeasibility);

      assert(row != NULL);
      SCIP_CALL( SCIPaddCut(scip, sol, row, FALSE, cutoff) );
      SCIP_CALL( SCIPresetConsAge(scip, cons) );
      if ( *cutoff )
         return SCIP_OKAY;
      (*separated) = TRUE;
   }

   return SCIP_OKAY;
}

/** this method creates a row for time point @p curtime which ensures the capacity restriction of the cumulative constraint */
static
SCIP_RETCODE createCapacityRestrictionIntvars(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   SCIP_SOL*             sol,                /**< primal CIP solution, NULL for current LP solution */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished,          /**< number of jobs that finished before curtime or at curtime */
   SCIP_Bool             lower               /**< shall cuts be created due to lower or upper bounds? */
   )
{
   SCIP_CONSDATA* consdata;
   char name[SCIP_MAXSTRLEN];
   SCIP_Bool infeasible;
   int lhs; /* left hand side of constraint */

   SCIP_VAR** activevars;
   SCIP_ROW* row;

   int v;

   assert(nstarted > nfinished);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(consdata->nvars > 0);


   SCIP_CALL( SCIPallocBufferArray(scip, &activevars, nstarted-nfinished) );

   SCIP_CALL( collectIntVars(scip, consdata, &activevars, startindices, curtime, nstarted, nfinished, lower, &lhs ) );

   if( lower )
   {
      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "lower(%d)", curtime);

      SCIP_CALL( SCIPcreateEmptyRowCons(scip, &row, SCIPconsGetHdlr(cons), name, (SCIP_Real) lhs, SCIPinfinity(scip),
            TRUE, FALSE, SCIPconsIsRemovable(cons)) );
   }
   else
   {
      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "upper(%d)", curtime);
      SCIP_CALL( SCIPcreateEmptyRowCons(scip, &row, SCIPconsGetHdlr(cons), name, -SCIPinfinity(scip), (SCIP_Real) lhs,
            TRUE, FALSE, SCIPconsIsRemovable(cons)) );
   }

   SCIP_CALL( SCIPcacheRowExtensions(scip, row) );

   for( v = 0; v < nstarted - nfinished; ++v )
   {
      SCIP_CALL( SCIPaddVarToRow(scip, row, activevars[v], 1.0) );
   }

   SCIP_CALL( SCIPflushRowExtensions(scip, row) );
   SCIPdebug( SCIP_CALL(SCIPprintRow(scip, row, NULL)) );

   SCIP_CALL( SCIPaddCut(scip, sol, row, TRUE, &infeasible) );
   assert( ! infeasible );

   SCIP_CALL( SCIPreleaseRow(scip, &row) );

   /* free buffers */
   SCIPfreeBufferArrayNull(scip, &activevars);

   return SCIP_OKAY;
}

/** checks constraint for violation, and adds it as a cut if possible */
static
SCIP_RETCODE separateConsOnIntegerVariables(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint to be separated */
   SCIP_SOL*             sol,                /**< primal CIP solution, NULL for current LP solution */
   SCIP_Bool             lower,              /**< shall cuts be created according to lower bounds? */
   SCIP_Bool*            separated           /**< pointer to store TRUE, if a cut was found */
   )
{

   SCIP_CONSDATA* consdata;

   int* starttimes;         /* stores when each job is starting */
   int* endtimes;           /* stores when each job ends */
   int* startindices;       /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;         /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   int nvars;               /* number of activities for this constraint */
   int freecapacity;        /* remaining capacity */
   int curtime;             /* point in time which we are just checking */
   int endindex;            /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int hmin;
   int hmax;
   int j;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   /* if no activities are associated with this cumulative then this constraint is redundant */
   if( nvars <= 1 )
      return SCIP_OKAY;

   assert(consdata->vars != NULL);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   SCIPdebugMessage("create sorted event points for cumulative constraint <%s> with %d jobs\n",
      SCIPconsGetName(cons), nvars);

   /* create event point arrays */
   createSelectedSortedEventpointsSol(scip, consdata, sol, starttimes, endtimes, startindices, endindices, &nvars, lower);

   /* now nvars might be smaller than before! */

   endindex = 0;
   freecapacity = consdata->capacity;
   hmin = consdata->hmin;
   hmax = consdata->hmax;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars; ++j )
   {
      curtime = starttimes[j];

      if( curtime >= hmax )
         break;

      /* remove the capacity requirements for all job which start at the curtime */
      subtractStartingJobDemands(consdata, curtime, starttimes, startindices, &freecapacity, &j, nvars);

      /* add the capacity requirments for all job which end at the curtime */
      addEndingJobDemands(consdata, curtime, endtimes, endindices, &freecapacity, &endindex, nvars);

      assert(freecapacity <= consdata->capacity);
      assert(endindex <= nvars);

      /* endindex - points to the next job which will finish */
      /* j - points to the last job that has been released */

      /* if free capacity is smaller than zero, then add rows to the LP */
      if( freecapacity < 0 && curtime >= hmin)
      {
         /* create capacity restriction row for current event point */
         SCIP_CALL( createCapacityRestrictionIntvars(scip, cons, sol, startindices, curtime, j+1, endindex, lower) );
         *separated = TRUE;
      }
   } /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   return SCIP_OKAY;
}

/**@} */


/**@name Presolving
 *
 * @{
 */

#ifndef NDEBUG
/** returns TRUE if all demands are smaller than the capacity of the cumulative constraint and if the total demand is
 *  correct
 */
static
SCIP_Bool checkDemands(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint to be checked */
   )
{
   SCIP_CONSDATA* consdata;
   int capacity;
   int nvars;
   int j;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   /* if no activities are associated with this cumulative then this constraint is not infeasible, return */
   if( nvars <= 1 )
      return TRUE;

   assert(consdata->vars != NULL);
   capacity = consdata->capacity;

   /* check each activity: if demand is larger than capacity the problem is infeasible */
   for ( j = 0; j < nvars; ++j )
   {
      if( consdata->demands[j] > capacity )
         return FALSE;
   }

   return TRUE;
}
#endif

/** delete constraint if it consists of at most one job
 *
 *  @todo this method needs to be adjusted w.r.t. effective horizon
 */
static
SCIP_RETCODE deleteTrivilCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to propagate */
   int*                  ndelconss,          /**< pointer to store the number of deleted constraints */
   SCIP_Bool*            cutoff              /**< pointer to store if the constraint is infeasible */
   )
{
   SCIP_CONSDATA* consdata;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   if( consdata->nvars == 0 )
   {
      SCIPdebugMessage("delete cumulative constraints <%s>\n", SCIPconsGetName(cons));

      SCIP_CALL( SCIPdelCons(scip, cons) );
      (*ndelconss)++;
   }
   else if( consdata->nvars == 1 )
   {
      if( consdata->demands[0] > consdata->capacity )
         (*cutoff) = TRUE;
      else
      {
         SCIPdebugMessage("delete cumulative constraints <%s>\n", SCIPconsGetName(cons));

         SCIP_CALL( SCIPdelCons(scip, cons) );
         (*ndelconss)++;
      }
   }

   return SCIP_OKAY;
}

/** remove jobs which have a duration or demand of zero (zero energy) or lay outside the efficient horizon [hmin, hmax);
 *  this is done in the SCIP_DECL_CONSINITPRE() callback
 */
static
SCIP_RETCODE removeIrrelevantJobs(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint to propagate */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR* var;
   int demand;
   int duration;
   int hmin;
   int hmax;
   int est;
   int lct;
   int j;

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   hmin = consdata->hmin;
   hmax = consdata->hmax;

   SCIPdebugMessage("check for irrelevant jobs within cumulative constraint <%s>[%d,%d)\n",
      SCIPconsGetName(cons), hmin, hmax);

   for( j = consdata->nvars-1; j >= 0; --j )
   {
      var = consdata->vars[j];
      demand = consdata->demands[j];
      duration = consdata->durations[j];

      /* earliest completion time (ect) and latest start time (lst) */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));
      lct = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var)) + duration;

      if( demand == 0 || duration == 0 )
      {
         /* jobs with zero demand or zero duration can be removed */
         SCIPdebugMessage("  remove variable <%s> due to zero %s\n",
            SCIPvarGetName(var), demand == 0 ? "demand" : "duration");

         /* remove variable form constraint */
         SCIP_CALL( consdataDeletePos(scip, consdata, cons, j) );
      }
      else if( est >= hmax || lct <= hmin )
      {
         SCIPdebugMessage("  remove variable <%s>[%d,%d] with duration <%d>\n",
            SCIPvarGetName(var), est, lct - duration, duration);

         /* delete variable at the given position */
         SCIP_CALL( consdataDeletePos(scip, consdata, cons, j) );

         /* for the statistic we count the number of jobs which are irrelevant */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nirrelevantjobs++ );
      }
   }

   return SCIP_OKAY;
}

/** adjust bounds of over sizeed job (the demand is larger than the capacity) */
static
SCIP_RETCODE adjustOversizedJobBounds(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   int                   pos,                /**< position of job in the consdata */
   int*                  nchgbds,            /**< pointer to store the number of changed bounds */
   int*                  naddconss,          /**< pointer to store the number of added constraints */
   SCIP_Bool*            cutoff              /**< pointer to store if a cutoff was detected */
   )
{
   SCIP_VAR* var;
   SCIP_Bool tightened;
   int duration;
   int ect;
   int lst;

   assert(scip != NULL);

   /* zero energy jobs should be removed already */
   assert(consdata->durations[pos] > 0);
   assert(consdata->demands[pos] > 0);

   var = consdata->vars[pos];
   assert(var != NULL);
   duration =  consdata->durations[pos];

   /* jobs with a demand greater than the the capacity have to moved outside the time interval [hmin,hmax) */
   SCIPdebugMessage("  variable <%s>: demand <%d> is larger than the capacity <%d>\n",
      SCIPvarGetName(var), consdata->demands[pos], consdata->capacity);

   /* earliest completion time (ect) and latest start time (lst) */
   ect = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var)) + duration;
   lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));

   /* the jobs has to have an overlap with the efficient horizon otherwise it would be already removed */
   if( ect - duration >= consdata->hmax || lst + duration <= consdata->hmin)
      return SCIP_OKAY;

   if( ect > consdata->hmin && lst < consdata->hmax )
   {
      /* the job will at least run partly in the time interval [hmin,hmax) this means the problem is infeasible */
      *cutoff = TRUE;
   }
   else if( lst < consdata->hmax )
   {
      /* move the latest start time of this job in such a way that it finishes before or at hmin */
      SCIP_CALL( SCIPtightenVarUb(scip, var, (SCIP_Real)(consdata->hmin - duration), TRUE, cutoff, &tightened) );
      assert(tightened);
      assert(!(*cutoff));
      (*nchgbds)++;
   }
   else if( ect > consdata->hmin )
   {
      /* move the earliest start time of this job in such a way that it starts after or at hmax */
      SCIP_CALL( SCIPtightenVarLb(scip, var, (SCIP_Real)(consdata->hmax), TRUE, cutoff, &tightened) );
      assert(tightened);
      assert(!(*cutoff));
      (*nchgbds)++;
   }
   else
   {
      /* this job can run before or after the time interval [hmin,hmax) thus we create a bound disjunction
       * constraint to ensure that it does not overlap with the time interval [hmin,hmax); that is:
       *
       * (var <= hmin - duration) /\ (var >= hmax)
       */
      SCIP_CONS* cons;

      SCIP_VAR* vartuple[2];
      SCIP_BOUNDTYPE boundtypetuple[2];
      SCIP_Real boundtuple[2];

      char name[SCIP_MAXSTRLEN];
      int leftbound;
      int rightbound;

      leftbound = consdata->hmin - duration;
      rightbound = consdata->hmax;

      /* allocate temporary memory for arrays */
      vartuple[0] = var;
      vartuple[1] = var;
      boundtuple[0] = (SCIP_Real)leftbound;
      boundtuple[1] = (SCIP_Real)rightbound;
      boundtypetuple[0] = SCIP_BOUNDTYPE_UPPER;
      boundtypetuple[1] = SCIP_BOUNDTYPE_LOWER;

      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "%s<=%d or %s >= %d",
         SCIPvarGetName(var), leftbound, SCIPvarGetName(var), rightbound);

      /* create and add bounddisjunction constraint */
      SCIP_CALL( SCIPcreateConsBounddisjunction(scip, &cons, name, 2, vartuple, boundtypetuple, boundtuple,
            TRUE, FALSE, TRUE, TRUE /*check*/, TRUE/*prop*/, FALSE, FALSE, FALSE, FALSE, FALSE) );

      SCIPdebugPrintCons(scip, cons, NULL);

      /* add and release the new constraint */
      SCIP_CALL( SCIPaddCons(scip, cons) );
      SCIP_CALL( SCIPreleaseCons(scip, &cons) );
      (*naddconss)++;
   }

   return SCIP_OKAY;
}

/** try to removed over sizeed jobs (the demand is larger than the capacity) */
static
SCIP_RETCODE removeOversizedJobs(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint */
   int*                  nchgbds,            /**< pointer to store the number of changed bounds */
   int*                  nchgcoefs,          /**< pointer to store the number of changed coefficient */
   int*                  naddconss,          /**< pointer to store the number of added constraints */
   SCIP_Bool*            cutoff              /**< pointer to store if a cutoff was detected */
   )
{
   SCIP_CONSDATA* consdata;
   int capacity;
   int j;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   /* if a cutoff was already detected just return */
   if( *cutoff )
      return SCIP_OKAY;

   capacity = consdata->capacity;

   for( j = consdata->nvars-1; j >= 0 && !(*cutoff);  --j )
   {
      if( consdata->demands[j] > capacity )
      {
         SCIP_CALL( adjustOversizedJobBounds(scip, consdata, j, nchgbds, naddconss, cutoff) );

         /* remove variable form constraint */
         SCIP_CALL( consdataDeletePos(scip, consdata, cons, j) );
         (*nchgcoefs)++;
      }
   }

   SCIPdebugMessage("cumulative constraint <%s> has %d jobs left, cutoff %u\n", SCIPconsGetName(cons), consdata->nvars, *cutoff);

   return SCIP_OKAY;
}

/** fix integer variable to upper bound if the rounding locks and the object coefficient are in favor of that */
static
SCIP_RETCODE fixIntegerVariableUb(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< integer variable to fix */
   SCIP_Bool             uplock,             /**< has thet start time variable a up lock */
   int*                  nfixedvars          /**< pointer to store the number fixed variables */
   )
{
   SCIP_Bool infeasible;
   SCIP_Bool tightened;
   SCIP_Bool roundable;

   /* if SCIP is in probing mode or repropagation we cannot perform this dual reductions since this dual reduction
    * would/could end in an implication which can lead to cutoff of the/all optimal solution
    */
   if( SCIPinProbing(scip) || SCIPinRepropagation(scip) )
      return SCIP_OKAY;

   /* rounding the variable to the upper bound is only a feasible dual reduction if the cumulative constraint
    * handler is the only one locking that variable up
    */
   assert(uplock == TRUE || uplock == FALSE);
   assert((int)TRUE == 1);
   assert((int)FALSE == 0);

   if( SCIPvarGetNLocksUp(var) > (int)(uplock) )
      return SCIP_OKAY;

   SCIP_CALL( varMayRoundUp(scip, var, &roundable) );

   /* rounding the integer variable up is only a valid dual reduction if the object coefficient is zero or negative
    * (the transformed problem is always a minimization problem)
    */
   if( !roundable )
      return SCIP_OKAY;

   SCIPdebugMessage("try fixing variable <%s>[%g,%g] to upper bound %g\n", SCIPvarGetName(var),
      SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), SCIPvarGetUbLocal(var));

   SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetUbLocal(var), &infeasible, &tightened) );
   assert(!infeasible);

   if( tightened )
   {
      SCIPdebugMessage("fix variable <%s> to upper bound %g\n", SCIPvarGetName(var), SCIPvarGetUbLocal(var));
      (*nfixedvars)++;
   }

   return SCIP_OKAY;
}

/** fix integer variable to lower bound if the rounding locks and the object coefficient are in favor of that */
static
SCIP_RETCODE fixIntegerVariableLb(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< integer variable to fix */
   SCIP_Bool             downlock,           /**< has the variable a down lock */
   int*                  nfixedvars          /**< pointer to store the number fixed variables */
   )
{
   SCIP_Bool infeasible;
   SCIP_Bool tightened;
   SCIP_Bool roundable;

   /* if SCIP is in probing mode or repropagation we cannot perform this dual reductions since this dual reduction
    * would/could end in an implication which can lead to cutoff of the/all optimal solution
    */
   if( SCIPinProbing(scip) || SCIPinRepropagation(scip) )
      return SCIP_OKAY;

   /* rounding the variable to the lower bound is only a feasible dual reduction if the cumulative constraint
    * handler is the only one locking that variable down
    */
   assert(downlock == TRUE || downlock == FALSE);
   assert((int)TRUE == 1);
   assert((int)FALSE == 0);

   if( SCIPvarGetNLocksDown(var) > (int)(downlock) )
      return SCIP_OKAY;

   SCIP_CALL( varMayRoundDown(scip, var, &roundable) );

   /* is it possible, to round variable down w.r.t. objective function? */
   if( !roundable )
      return SCIP_OKAY;

   SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetLbLocal(var), &infeasible, &tightened) );
   assert(!infeasible);

   if( tightened )
   {
      SCIPdebugMessage("fix variable <%s> to lower bound %g\n", SCIPvarGetName(var), SCIPvarGetLbLocal(var));
      (*nfixedvars)++;
   }

   return SCIP_OKAY;
}

/** normalize cumulative condition */
static
SCIP_RETCODE normalizeCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int*                  capacity,           /**< pointer to store the changed cumulative capacity */
   int*                  nchgcoefs,          /**< pointer to count total number of changed coefficients */
   int*                  nchgsides           /**< pointer to count number of side changes */
   )
{  /*lint --e{715}*/
   SCIP_Longint gcd;
   int mindemand1;
   int mindemand2;
   int v;

   if( *capacity == 1 || nvars <= 1 )
      return SCIP_OKAY;

   assert(demands[nvars-1] <= *capacity);
   assert(demands[nvars-2] <= *capacity);

   gcd = (SCIP_Longint)demands[nvars-1];
   mindemand1 = MIN(demands[nvars-1], demands[nvars-2]);
   mindemand2 = MAX(demands[nvars-1], demands[nvars-2]);

   for( v = nvars-2; v >= 0 && (gcd >= 2 || mindemand1 + mindemand2 > *capacity); --v )
   {
      assert(mindemand1 <= mindemand2);
      assert(demands[v] <= *capacity);

      gcd = SCIPcalcGreComDiv(gcd, (SCIP_Longint)demands[v]);

      if( mindemand1 > demands[v] )
      {
         mindemand2 = mindemand1;
         mindemand1 = demands[v];
      }
      else if( mindemand2 > demands[v] )
         mindemand2 = demands[v];
   }

   if( mindemand1 + mindemand2 > *capacity )
   {
      SCIPdebugMessage("update cumulative condition (%d + %d > %d) to unary cumulative condition\n", mindemand1, mindemand2, *capacity);

      for( v = 0; v < nvars; ++v )
         demands[v] = 1;

      (*capacity) = 1;

      (*nchgcoefs) += nvars;
      (*nchgsides)++;
   }
   else if( gcd >= 2 )
   {
      SCIPdebugMessage("cumulative condition: dividing demands by %" SCIP_LONGINT_FORMAT "\n", gcd);

      for( v = 0; v < nvars; ++v )
         demands[v] /= gcd;

      (*capacity) /= gcd;

      (*nchgcoefs) += nvars;
      (*nchgsides)++;
   }

   return SCIP_OKAY;
}

/** divides demands by their greatest common divisor and divides capacity by the same value, rounding down the result;
 *  in case the the smallest demands add up to more than the capacity we reductions all demands to one as well as the
 *  capacity since in that case none of the jobs can run in parallel
 */
static
SCIP_RETCODE normalizeDemands(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  nchgcoefs,          /**< pointer to count total number of changed coefficients */
   int*                  nchgsides           /**< pointer to count number of side changes */
   )
{
   SCIP_CONSDATA* consdata;
   int capacity;

   assert(nchgcoefs != NULL);
   assert(nchgsides != NULL);
   assert(!SCIPconsIsModifiable(cons));

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   if( consdata->normalized )
      return SCIP_OKAY;

   capacity = consdata->capacity;

   /**@todo sort items w.r.t. the demands, because we can stop earlier if the smaller weights are evaluated first */

   SCIP_CALL( normalizeCumulativeCondition(scip, consdata->nvars, consdata->vars, consdata->durations,
         consdata->demands, &consdata->capacity, nchgcoefs, nchgsides) );

   consdata->normalized = TRUE;

   if( capacity > consdata->capacity )
      consdata->varbounds = FALSE;

   return SCIP_OKAY;
}

/** computes for the given cumulative condition the effective horizon */
static
SCIP_RETCODE computeEffectiveHorizonCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int*                  hmin,               /**< pointer to store the left bound of the effective horizon */
   int*                  hmax,               /**< pointer to store the right bound of the effective horizon */
   int*                  split               /**< point were the cumulative condition can be split */
   )
{
   SCIP_PROFILE* profile;

   /* create empty resource profile with infinity resource capacity */
   SCIP_CALL( SCIPprofileCreate(&profile, INT_MAX) );

   /* create worst case resource profile */
   SCIP_CALL( SCIPcreateWorstCaseProfile(scip, profile, nvars, vars, durations, demands) );

   /* print resource profile in if SCIP_DEBUG is defined */
   SCIPdebug( SCIPprofilePrint(profile, SCIPgetMessagehdlr(scip), NULL) );

   /* computes the first time point where the resource capacity can be violated */
   (*hmin) = SCIPcomputeHmin(scip, profile, capacity);

   /* computes the first time point where the resource capacity is satisfied for sure */
   (*hmax) = SCIPcomputeHmax(scip, profile, capacity);

   (*split) = (*hmax);

   if( *hmin < *hmax && !SCIPinRepropagation(scip) )
   {
      int* timepoints;
      int* loads;
      int ntimepoints;
      int t;

      /* If SCIP is repropagating the root node, it is not possible to decompose the constraints. This is the case since
       * the conflict analysis stores the constraint pointer for bound changes made by this constraint. These pointer
       * are used during the resolve propagation phase to explain bound changes. If we would decompose certain jobs into
       * a new cumulative constraint, the "old" pointer is not valid. More precise, the "old" constraint is not able to
       * explain the certain "old" bound changes
       */

      /* search for time points */
      ntimepoints = SCIPprofileGetNTimepoints(profile);
      timepoints = SCIPprofileGetTimepoints(profile);
      loads = SCIPprofileGetLoads(profile);

      /* check if there exist a time point within the effective horizon [hmin,hmax) such that the capacity is not exceed w.r.t. worst case profile */
      for( t = 0; t < ntimepoints; ++t )
      {
         /* ignore all time points before the effective horizon */
         if( timepoints[t] <= *hmin )
            continue;

         /* ignore all time points after the effective horizon */
         if( timepoints[t] >= *hmax )
            break;

         /* check if the current time point does not exceed the capacity w.r.t. worst case resource profile; if so we
          * can split the cumulative constraint into two cumulative constraints
          */
         if( loads[t] <= capacity )
         {
            (*split) = timepoints[t];
            break;
         }
      }
   }

   /* free worst case profile */
   SCIPprofileFree(&profile);

   return SCIP_OKAY;
}

/** creates and adds a cumulative constraint */
static
SCIP_RETCODE createConsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   const char*           name,               /**< name of constraint */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool             initial,            /**< should the LP relaxation of constraint be in the initial LP?
                                              *   Usually set to TRUE. Set to FALSE for 'lazy constraints'. */
   SCIP_Bool             separate,           /**< should the constraint be separated during LP processing?
                                              *   Usually set to TRUE. */
   SCIP_Bool             enforce,            /**< should the constraint be enforced during node processing?
                                              *   TRUE for model constraints, FALSE for additional, redundant constraints. */
   SCIP_Bool             check,              /**< should the constraint be checked for feasibility?
                                              *   TRUE for model constraints, FALSE for additional, redundant constraints. */
   SCIP_Bool             propagate,          /**< should the constraint be propagated during node processing?
                                              *   Usually set to TRUE. */
   SCIP_Bool             local,              /**< is constraint only valid locally?
                                              *   Usually set to FALSE. Has to be set to TRUE, e.g., for branching constraints. */
   SCIP_Bool             modifiable,         /**< is constraint modifiable (subject to column generation)?
                                              *   Usually set to FALSE. In column generation applications, set to TRUE if pricing
                                              *   adds coefficients to this constraint. */
   SCIP_Bool             dynamic,            /**< is constraint subject to aging?
                                              *   Usually set to FALSE. Set to TRUE for own cuts which
                                              *   are seperated as constraints. */
   SCIP_Bool             removable,          /**< should the relaxation be removed from the LP due to aging or cleanup?
                                              *   Usually set to FALSE. Set to TRUE for 'lazy constraints' and 'user cuts'. */
   SCIP_Bool             stickingatnode      /**< should the constraint always be kept at the node where it was added, even
                                              *   if it may be moved to a more global node?
                                              *   Usually set to FALSE. Set to TRUE to for constraints that represent node data. */
   )
{
   SCIP_CONS* cons;

   /* creates cumulative constraint and adds it to problem */
   SCIP_CALL( SCIPcreateConsCumulative(scip, &cons, name, nvars, vars, durations, demands, capacity,
         initial, separate, enforce, check, propagate, local, modifiable, dynamic, removable, stickingatnode) );

   /* adjust the effective time horizon of the new constraint */
   SCIP_CALL( SCIPsetHminCumulative(scip, cons, hmin) );
   SCIP_CALL( SCIPsetHmaxCumulative(scip, cons, hmax) );

   /* add and release new cumulative constraint */
   SCIP_CALL( SCIPaddCons(scip, cons) );
   SCIP_CALL( SCIPreleaseCons(scip, &cons) );

   return SCIP_OKAY;
}

/** computes the effective horizon and checks if the constraint can be decompsed */
static
SCIP_RETCODE computeEffectiveHorizon(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  ndelconss,          /**< pointer to store the number of deleted constraints */
   int*                  naddconss,          /**< pointer to store the number of added constraints */
   int*                  nchgsides           /**< pointer to store the number of changed sides */
   )
{
   SCIP_CONSDATA* consdata;
   int hmin;
   int hmax;
   int split;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   if( consdata->nvars <= 1 )
      return SCIP_OKAY;

   SCIP_CALL( computeEffectiveHorizonCumulativeCondition(scip, consdata->nvars, consdata->vars,
         consdata->durations, consdata->demands, consdata->capacity, &hmin, &hmax, &split) );

   /* check if this time point improves the effective horizon */
   if( consdata->hmin < hmin )
   {
      SCIPdebugMessage("cumulative constraint <%s> adjust hmin <%d> -> <%d>\n", SCIPconsGetName(cons), consdata->hmin, hmin);

      consdata->hmin = hmin;
      (*nchgsides)++;
   }

   /* check if this time point improves the effective horizon */
   if( consdata->hmax > hmax )
   {
      SCIPdebugMessage("cumulative constraint <%s> adjust hmax <%d> -> <%d>\n", SCIPconsGetName(cons), consdata->hmax,  hmax);
      consdata->hmax = hmax;
      (*nchgsides)++;
   }

   /* check if the constraint is redundant */
   if( consdata->hmax <= consdata->hmin )
   {
      SCIPdebugMessage("constraint <%s> is redundant since hmax(%d) <= hmin(%d)\n",
         SCIPconsGetName(cons), consdata->hmax, consdata->hmin);

      SCIP_CALL( SCIPdelCons(scip, cons) );
      (*ndelconss)++;
   }
   else if( consdata->hmin < split && split < consdata->hmax )
   {
      char name[SCIP_MAXSTRLEN];
      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "(%s)'", SCIPconsGetName(cons));

      SCIPdebugMessage("split cumulative constraint <%s>[%d,%d) with %d jobs at time point %d\n",
         SCIPconsGetName(cons), consdata->hmin, consdata->hmax, consdata->nvars, split);

      assert(split < consdata->hmax);

      /* creates cumulative constraint and adds it to problem */
      SCIP_CALL( createConsCumulative(scip, name, consdata->nvars, consdata->vars,
            consdata->durations, consdata->demands, consdata->capacity, split, consdata->hmax,
            SCIPconsIsInitial(cons), SCIPconsIsSeparated(cons), SCIPconsIsEnforced(cons), SCIPconsIsChecked(cons), SCIPconsIsPropagated(cons),
            SCIPconsIsLocal(cons), SCIPconsIsModifiable(cons), SCIPconsIsDynamic(cons), SCIPconsIsRemovable(cons), SCIPconsIsStickingAtNode(cons)) );

      /* adjust the effective time horizon of the constraint */
      consdata->hmax = split;

      assert(consdata->hmin < consdata->hmax);

      /* for the statistic we count the number of time we decompose a cumulative constraint */
      SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndecomps++ );
      (*naddconss)++;
   }

   return SCIP_OKAY;
}


/** presolve cumulative condition w.r.t. the earlier start times (est) and the hmin of the effective horizon
 *
 *  (1) If the latest completion time (lct) of a job is smaller or equal than hmin, the corresponding job can be removed
 *      form the constraint. This is the case since it cannot effect any assignment within the effective horizon
 *
 *  (2) If the latest start time (lst) of a job is smaller or equal than hmin it follows that the this jobs can run
 *      before the effective horizon or it overlaps with the effective horizon such that hmin in included. Hence, the
 *      down-lock of the corresponding start time variable can be removed.
 *
 *  (3) If the earlier completion time (ect) of a job is smaller or equal than hmin, the cumulative is the only one
 *      locking the corresponding variable down, and the objective coefficient of the start time variable is not
 *      negative, than the job can be dual fixed to its earlier start time (est).
 *
 *  (4) If the earlier start time (est) of job is smaller than the hmin, the cumulative is the only one locking the
 *      corresponding variable down, and the objective coefficient of the start time variable is not negative, than
 *      removing the values {est+1,...,hmin} form variable domain is dual feasible.
 *
 *  (5) If the earlier start time (est) of job is smaller than the smallest earlier completion times of all other jobs
 *      (lets denote this with minect), the cumulative is the only one locking the corresponding variable down, and the
 *      objective coefficient of the start time variable is not negative, than removing the values {est+1,...,minect-1}
 *      form variable domain is dual feasible.
 *
 *  @note That method does not remove any variable form the arrays. It only marks the variables which are irrelevant for
 *        the cumulative condition; The deletion has to be done later.
 */
static
SCIP_RETCODE presolveConsEst(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            downlocks,          /**< array to store if the variable has a down lock, or NULL */
   SCIP_Bool*            uplocks,            /**< array to store if the variable has an up lock, or NULL */
   SCIP_CONS*            cons,               /**< underlying constraint, or NULL */
   SCIP_Bool*            irrelevants,        /**< array mark those variables which are irrelevant for the cumulative condition */
   int*                  nfixedvars,         /**< pointer to store the number of fixed variables */
   int*                  nchgsides,          /**< pointer to store the number of changed sides */
   SCIP_Bool*            cutoff              /**< buffer to store whether a cutoff is detected */
   )
{
   SCIP_Real* downimpllbs;
   SCIP_Real* downimplubs;
   SCIP_Real* downproplbs;
   SCIP_Real* downpropubs;
   SCIP_Real* upimpllbs;
   SCIP_Real* upimplubs;
   SCIP_Real* upproplbs;
   SCIP_Real* uppropubs;

   int firstminect;
   int secondminect;
   int v;

   /* get temporary memory for storing probing results needed for step (4) and (5) */
   SCIP_CALL( SCIPallocBufferArray(scip, &downimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downpropubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &uppropubs, nvars) );

   assert(scip != NULL);
   assert(nvars > 1);
   assert(cons != NULL);

   SCIPdebugMessage("check for irrelevant variable for cumulative condition (hmin %d) w.r.t. earlier start time\n", hmin);

   firstminect = INT_MAX;
   secondminect = INT_MAX;

   /* compute the two smallest earlier completion times; which are needed for step (5) */
   for( v = 0; v < nvars; ++v )
   {
      int ect;

      ect = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(vars[v])) + durations[v];

      if( ect < firstminect )
      {
         secondminect = firstminect;
         firstminect = ect;
      }
      else if( ect < secondminect )
         secondminect = ect;
   }

   /* loop over all jobs and check if one of the 5 reductions can be applied */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      int duration;

      int alternativelb;
      int minect;
      int est;
      int ect;
      int lst;
      int lct;

      var = vars[v];
      assert(var != NULL);

      duration = durations[v];
      assert(duration > 0);

      /* collect earlier start time (est), earlier completion time (ect), latest start time (lst), and latest completion
       * time (lct)
       */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));
      ect = est + duration;
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));
      lct = lst + duration;

      /* compute the earliest completion time of all remaining jobs */
      if( ect == firstminect )
         minect = secondminect;
      else
         minect = firstminect;

      /* compute potential alternative lower bound (step (4) and (5)) */
      alternativelb = MAX(hmin+1, minect);
      alternativelb = MIN(alternativelb, hmax);

      if( lct <= hmin )
      {
         /* (1) check if the job runs completely before the effective horizon; if so the job can be removed form the
          * cumulative condition
          */
         SCIPdebugMessage("  variable <%s>[%g,%g] with duration <%d> is irrelevant\n",
            SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration);

         /* mark variable to be irrelevant */
         irrelevants[v] = TRUE;

         /* for the statistic we count the number of jobs which are irrelevant */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nirrelevantjobs++ );
      }
      else if( lst <= hmin && SCIPconsIsChecked(cons) )
      {
         /* (2) check if the jobs overlaps with the time point hmin if it overlaps at all with the effective horizon; if
          * so the down lock can be omitted
          */

         assert(downlocks != NULL);
         assert(uplocks != NULL);

         if( !uplocks[v] )
         {
            /* the variables has no up lock and we can also remove the down lock;
             * => lst <= hmin and ect >= hmax
             * => remove job and reduce capacity by the demand of that job
             *
             * We mark the job to be deletable. The removement together with the capacity reducion is done later
             */

            SCIPdebugMessage("  variables <%s>[%d,%d] (duration <%d>) is irrelevant due to no up lock\n",
               SCIPvarGetName(var), ect - duration, lst, duration);

            /* mark variable to be irrelevant */
            irrelevants[v] = TRUE;

            /* for the statistic we count the number of jobs which always run during the effective horizon */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nalwaysruns++ );
         }

         if( downlocks[v] )
         {
            SCIPdebugMessage("  remove down lock of variable <%s>[%g,%g] with duration <%d>\n",
               SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration);

            SCIP_CALL( SCIPunlockVarCons(scip, var, cons, TRUE, FALSE) );
            downlocks[v] = FALSE;
            (*nchgsides)++;

            /* for the statistic we count the number of removed locks */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nremovedlocks++ );
         }
      }
      else if( ect <= hmin )
      {
         /* (3) check if the job can finish before the effective horizon starts; if so and the job can be fixed to its
          * earliest start time (which implies that it finishes before the effective horizon starts), the job can be
          * removed form the cumulative condition after it was fixed to its earliest start time
          */

         /* job can be removed from the constraint only if the integer start time variable can be fixed to its lower
          * bound;
          */
         if( downlocks != NULL && SCIPconsIsChecked(cons) )
         {
            /* fix integer start time variable if possible to it lower bound */
            SCIP_CALL( fixIntegerVariableLb(scip, var, downlocks[v], nfixedvars) );
         }

         if( SCIPvarGetLbGlobal(var) + 0.5 > SCIPvarGetUbGlobal(var) )
         {
            SCIPdebugMessage("  variable <%s>[%d,%d] with duration <%d> is irrelevant due to dual fixing wrt EST\n",
               SCIPvarGetName(var), ect - duration, lst, duration);

            /* after fixing the start time variable to its lower bound, the (new) earliest completion time should be smaller or equal ti hmin */
            assert(SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var)) + duration <= hmin);

            /* mark variable to be irrelevant */
            irrelevants[v] = TRUE;

            /* for the statistic we count the number of jobs which are dual fixed */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualfixs++ );
         }
      }
      else if( est < lst && est < alternativelb && SCIPconsIsChecked(cons) )
      {
         assert(downlocks != NULL);

         /* check step (4) and (5) */

         /* check if the cumulative constraint is the only one looking this variable down and if the objective function
          * is in favor of rounding the variable down
          */
         if( SCIPvarGetNLocksDown(var) == (int)(downlocks[v]) )
         {
            SCIP_Bool roundable;

            SCIP_CALL( varMayRoundDown(scip, var, &roundable) );

            if( roundable )
            {
               if( alternativelb > lst )
               {
                  SCIP_Bool infeasible;
                  SCIP_Bool fixed;

                  SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetLbLocal(var), &infeasible, &fixed) );
                  assert(!infeasible);
                  assert(fixed);

                  (*nfixedvars)++;

                  /* for the statistic we count the number of jobs which are dual fixed due the information of all cumulative
                   * constraints
                   */
                  SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualbranchs++ );
               }
               else
               {
                  SCIP_Bool success;

                  /* In the current version SCIP, variable domains are single intervals. Meaning that domain holes or not
                   * representable. To retrieve a potential dual reduction we using probing to check both branches. If one in
                   * infeasible we can apply the dual reduction; otherwise we do nothing
                   */
                  SCIP_CALL( applyProbingVar(scip, vars, nvars, v, (SCIP_Real) est, (SCIP_Real) alternativelb,
                        downimpllbs, downimplubs, downproplbs, downpropubs, upimpllbs, upimplubs, upproplbs, uppropubs,
                        nfixedvars, &success, cutoff) );

                  if( success )
                  {
                     SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualbranchs++ );
                  }
               }
            }
         }
      }

      SCIPdebugMessage("********* check variable <%s>[%g,%g] with duration <%d> (hmin %d)\n",
         SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration, hmin);
   }

   /* free temporary memory */
   SCIPfreeBufferArray(scip, &uppropubs);
   SCIPfreeBufferArray(scip, &upproplbs);
   SCIPfreeBufferArray(scip, &upimplubs);
   SCIPfreeBufferArray(scip, &upimpllbs);
   SCIPfreeBufferArray(scip, &downpropubs);
   SCIPfreeBufferArray(scip, &downproplbs);
   SCIPfreeBufferArray(scip, &downimplubs);
   SCIPfreeBufferArray(scip, &downimpllbs);

   return SCIP_OKAY;
}

/** presolve cumulative condition w.r.t. the latest completion times (lct) and the hmax of the effective horizon
 *
 *  (1) If the earliest start time (est) of a job is larger or equal than hmax, the corresponding job can be removed
 *      form the constraint. This is the case since it cannot effect any assignment within the effective horizon
 *
 *  (2) If the earliest completion time (ect) of a job is larger or equal than hmax it follows that the this jobs can run
 *      before the effective horizon or it overlaps with the effective horizon such that hmax in included. Hence, the
 *      up-lock of the corresponding start time variable can be removed.
 *
 *  (3) If the latest start time (lst) of a job is larger or equal than hmax, the cumulative is the only one
 *      locking the corresponding variable up, and the objective coefficient of the start time variable is not
 *      positive, than the job can be dual fixed to its latest start time (lst).
 *
 *  (4) If the latest completion time (lct) of job is larger than the hmax, the cumulative is the only one locking the
 *      corresponding variable up, and the objective coefficient of the start time variable is not positive, than
 *      removing the values {hmax - p_j, ..., lst-1} form variable domain is dual feasible (p_j is the processing time
 *      of the corresponding job).

 *  (5) If the latest completion time (lct) of job is smaller than the largerst latest start time of all other jobs
 *      (lets denote this with maxlst), the cumulative is the only one locking the corresponding variable up, and the
 *      objective coefficient of the start time variable is not positive, than removing the values {maxlst - p_j + 1,
 *      ..., lst-1} form variable domain is dual feasible (p_j is the processing time of the corresponding job).
 *
 *  @note That method does not remove any variable form the arrays. It only marks the variables which are irrelevant for
 *        the cumulative condition; The deletion has to be done later.
 */
static
SCIP_RETCODE presolveConsLct(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            downlocks,          /**< array to store if the variable has a down lock, or NULL */
   SCIP_Bool*            uplocks,            /**< array to store if the variable has an up lock, or NULL */
   SCIP_CONS*            cons,               /**< underlying constraint, or NULL */
   SCIP_Bool*            irrelevants,        /**< array mark those variables which are irrelevant for the cumulative condition */
   int*                  nfixedvars,         /**< pointer to counter which is increased by the number of deduced variable fixations */
   int*                  nchgsides,          /**< pointer to store the number of changed sides */
   SCIP_Bool*            cutoff              /**< buffer to store whether a cutoff is detected */
   )
{
   SCIP_Real* downimpllbs;
   SCIP_Real* downimplubs;
   SCIP_Real* downproplbs;
   SCIP_Real* downpropubs;
   SCIP_Real* upimpllbs;
   SCIP_Real* upimplubs;
   SCIP_Real* upproplbs;
   SCIP_Real* uppropubs;

   int firstmaxlst;
   int secondmaxlst;
   int v;

   /* get temporary memory for storing probing results needed for step (4) and (5) */
   SCIP_CALL( SCIPallocBufferArray(scip, &downimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &downpropubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimpllbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upimplubs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &upproplbs, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &uppropubs, nvars) );

   assert(scip != NULL);
   assert(nvars > 1);
   assert(cons != NULL);

   SCIPdebugMessage("check for irrelevant variable for cumulative condition (hmax %d) w.r.t. latest completion time\n", hmax);

   firstmaxlst = INT_MIN;
   secondmaxlst = INT_MIN;

   /* compute the two largest latest start times; which are needed for step (5) */
   for( v = 0; v < nvars; ++v )
   {
      int lst;

      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(vars[v]));

      if( lst > firstmaxlst )
      {
         secondmaxlst = firstmaxlst;
         firstmaxlst = lst;
      }
      else if( lst > secondmaxlst )
         secondmaxlst = lst;
   }

   /* loop over all jobs and check if one of the 5 reductions can be applied */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      int duration;

      int alternativeub;
      int maxlst;
      int est;
      int ect;
      int lst;

      var = vars[v];
      assert(var != NULL);

      duration = durations[v];
      assert(duration > 0);

      /* collect earlier start time (est), earlier completion time (ect), latest start time (lst), and latest completion
       * time (lct)
       */
      est = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));
      ect = est + duration;
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));

      /* compute the latest start time of all remaining jobs */
      if( lst == firstmaxlst )
         maxlst = secondmaxlst;
      else
         maxlst = firstmaxlst;

      /* compute potential alternative upper bound (step (4) and (5)) */
      alternativeub = MIN(hmax - 1, maxlst)  - duration;
      alternativeub = MAX(alternativeub, hmin);

      if( est >= hmax )
      {
         /* (1) check if the job runs completely after the effective horizon; if so the job can be removed form the
          * cumulative condition
          */
         SCIPdebugMessage("  variable <%s>[%g,%g] with duration <%d> is irrelevant\n",
            SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration);

         /* mark variable to be irrelevant */
         irrelevants[v] = TRUE;

         /* for the statistic we count the number of jobs which are irrelevant */
         SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nirrelevantjobs++ );
      }
      else if( ect >= hmax && SCIPconsIsChecked(cons) )
      {
         assert(downlocks != NULL);
         assert(uplocks != NULL);

         /* (2) check if the jobs overlaps with the time point hmax if it overlaps at all with the effective horizon; if
          * so the up lock can be omitted
          */

         if( !downlocks[v] )
         {
            /* the variables has no down lock and we can also remove the up lock;
             * => lst <= hmin and ect >= hmax
             * => remove job and reduce capacity by the demand of that job
             */
            SCIPdebugMessage("  variables <%s>[%d,%d] with duration <%d> is irrelevant due to no down lock\n",
               SCIPvarGetName(var), est, lst, duration);

            /* mark variable to be irrelevant */
            irrelevants[v] = TRUE;

            /* for the statistic we count the number of jobs which always run during the effective horizon */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nalwaysruns++ );
         }

         if( uplocks[v] )
         {
            SCIPdebugMessage("  remove up lock of variable <%s>[%g,%g] with duration <%d>\n",
               SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), duration);

            SCIP_CALL( SCIPunlockVarCons(scip, var, cons, FALSE, TRUE) );
            uplocks[v] = FALSE;
            (*nchgsides)++;

            /* for the statistic we count the number of removed locks */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->nremovedlocks++ );
         }
      }
      else if( lst >= hmax )
      {
         /* (3) check if the job can start after the effective horizon finishes; if so and the job can be fixed to its
          * latest start time (which implies that it starts after the effective horizon finishes), the job can be
          * removed form the cumulative condition after it was fixed to its latest start time
          */

         /* job can be removed from the constraint only if the integer start time variable can be fixed to its upper
          * bound
          */
         if( uplocks != NULL && SCIPconsIsChecked(cons) )
         {
            /* fix integer start time variable if possible to its upper bound */
            SCIP_CALL( fixIntegerVariableUb(scip, var, uplocks[v], nfixedvars) );
         }

         if( SCIPvarGetLbGlobal(var) + 0.5 > SCIPvarGetUbGlobal(var) )
         {
            SCIPdebugMessage("  variable <%s>[%d,%d] with duration <%d> is irrelevant due to dual fixing wrt LCT\n",
               SCIPvarGetName(var), est, lst, duration);

            /* after fixing the start time variable to its upper bound, the (new) latest start time should be greather or equal ti hmax */
            assert(SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var)) >= hmax);

            /* mark variable to be irrelevant */
            irrelevants[v] = TRUE;

            /* for the statistic we count the number of jobs which are dual fixed */
            SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualfixs++ );
         }
      }
      else if( est < lst && lst > alternativeub && SCIPconsIsChecked(cons) )
      {
         assert(uplocks != NULL);

         /* check step (4) and (5) */

         /* check if the cumulative constraint is the only one looking this variable down and if the objective function
          * is in favor of rounding the variable down
          */
         if( SCIPvarGetNLocksUp(var) == (int)(uplocks[v]) )
         {
            SCIP_Bool roundable;

            SCIP_CALL( varMayRoundUp(scip, var, &roundable) );

            if( roundable )
            {
               if( alternativeub < est )
               {
                  SCIP_Bool infeasible;
                  SCIP_Bool fixed;

                  SCIP_CALL( SCIPfixVar(scip, var, SCIPvarGetUbLocal(var), &infeasible, &fixed) );
                  assert(!infeasible);
                  assert(fixed);

                  (*nfixedvars)++;

                  /* for the statistic we count the number of jobs which are dual fixed due the information of all cumulative
                   * constraints
                   */
                  SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualbranchs++ );
               }
               else
               {
                  SCIP_Bool success;

                  /* In the current version SCIP, variable domains are single intervals. Meaning that domain holes or not
                   * representable. To retrieve a potential dual reduction we using probing to check both branches. If one
                   * in infeasible we can apply the dual reduction; otherwise we do nothing
                   */
                  SCIP_CALL( applyProbingVar(scip, vars, nvars, v, (SCIP_Real) alternativeub, (SCIP_Real) lst,
                        downimpllbs, downimplubs, downproplbs, downpropubs, upimpllbs, upimplubs, upproplbs, uppropubs,
                        nfixedvars, &success, cutoff) );

                  if( success )
                  {
                     SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->ndualbranchs++ );
                  }
               }
            }
         }
      }
   }

   /* free temporary memory */
   SCIPfreeBufferArray(scip, &uppropubs);
   SCIPfreeBufferArray(scip, &upproplbs);
   SCIPfreeBufferArray(scip, &upimplubs);
   SCIPfreeBufferArray(scip, &upimpllbs);
   SCIPfreeBufferArray(scip, &downpropubs);
   SCIPfreeBufferArray(scip, &downproplbs);
   SCIPfreeBufferArray(scip, &downimplubs);
   SCIPfreeBufferArray(scip, &downimpllbs);

   return SCIP_OKAY;
}

/** presolve cumulative constraint w.r.t. the boundary of the effective horizon */
static
SCIP_RETCODE presolveConsEffectiveHorizon(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  nfixedvars,         /**< pointer to store the number of fixed variables */
   int*                  nchgcoefs,          /**< pointer to store the number of changed coefficients */
   int*                  nchgsides,          /**< pointer to store the number of changed sides */
   SCIP_Bool*            cutoff              /**< pointer to store if a cutoff was detected */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_Bool* irrelevants;
   int nvars;
   int v;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(!(*cutoff));

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   if( nvars <= 1 )
      return SCIP_OKAY;

   SCIP_CALL( SCIPallocBufferArray(scip, &irrelevants, nvars) );
   BMSclearMemoryArray(irrelevants, nvars);

   /* presolve constraint form the earlier start time point of view */
   SCIP_CALL( presolveConsEst(scip, nvars, consdata->vars, consdata->durations,
         consdata->hmin, consdata->hmax, consdata->downlocks, consdata->uplocks, cons,
         irrelevants, nfixedvars, nchgsides, cutoff) );

   /* presolve constraint form the latest completion time point of view */
   SCIP_CALL( presolveConsLct(scip, nvars, consdata->vars, consdata->durations,
         consdata->hmin, consdata->hmax, consdata->downlocks, consdata->uplocks, cons,
         irrelevants, nfixedvars, nchgsides, cutoff) );

   /* remove variables from the cumulative constraint which are marked to be deleted; we need to that in the reverse
    * order to ensure a correct behaviour
    */
   for( v = nvars-1; v >= 0; --v )
   {
      if( irrelevants[v] )
      {
         SCIP_VAR* var;
         int ect;
         int lst;

         var = consdata->vars[v];
         assert(var != NULL);

         ect = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var)) + consdata->durations[v];
         lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));

         /* check if the jobs runs completely during the effective horizon */
         if( lst <= consdata->hmin && ect >= consdata->hmax )
         {
            if( consdata->capacity < consdata->demands[v] )
            {
               *cutoff = TRUE;
               break;
            }

            consdata->capacity -= consdata->demands[v];
            consdata->varbounds = FALSE;
         }

         SCIP_CALL( consdataDeletePos(scip, consdata, cons, v) );
         (*nchgcoefs)++;
      }
   }

   SCIPfreeBufferArray(scip, &irrelevants);

   return SCIP_OKAY;
}

/** stores all demands which are smaller than the capacity of those jobs that are running at 'curtime' */
static
SCIP_RETCODE collectDemands(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSDATA*        consdata,           /**< constraint data */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished,          /**< number of jobs that finished before curtime or at curtime */
   SCIP_Longint**        demands,            /**< pointer to array storing the demands */
   int*                  ndemands            /**< pointer to store the number of different demands */
   )
{
   int startindex;
   int ncountedvars;

   assert(demands != NULL);
   assert(ndemands != NULL);

   ncountedvars = 0;
   startindex = nstarted - 1;

   *ndemands = 0;

   /* search for the (nstarted - nfinished) jobs which are active at curtime */
   while( nstarted - nfinished > ncountedvars )
   {
      SCIP_VAR* var;
      int endtime;
      int varidx;

      /* collect job information */
      varidx = startindices[startindex];
      assert(varidx >= 0 && varidx < consdata->nvars);

      var = consdata->vars[varidx];
      assert(var != NULL);

      endtime = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var)) + consdata->durations[varidx];

      /* check the end time of this job is larger than the curtime; in this case the job is still running */
      if( endtime > curtime )
      {
         if( consdata->demands[varidx] < consdata->capacity )
         {
            (*demands)[*ndemands] = consdata->demands[varidx];
            (*ndemands)++;
         }
         ncountedvars++;
      }

      startindex--;
   }

   return SCIP_OKAY;
}

/** this method creates a row for time point curtime which insures the capacity restriction of the cumulative
 *  constraint
 */
static
SCIP_RETCODE getHighestCapacityUsage(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to be checked */
   int*                  startindices,       /**< permutation with rspect to the start times */
   int                   curtime,            /**< current point in time */
   int                   nstarted,           /**< number of jobs that start before the curtime or at curtime */
   int                   nfinished,          /**< number of jobs that finished before curtime or at curtime */
   int*                  bestcapacity        /**< pointer to store the maximum possible capacity usage */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_Longint* demands;
   SCIP_Real* profits;
   int* items;
   int ndemands;
   SCIP_Bool success;
   SCIP_Real solval;
   int j;
   assert(nstarted > nfinished);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(consdata->nvars > 0);
   assert(consdata->capacity > 0);

   SCIP_CALL( SCIPallocBufferArray(scip, &demands, consdata->nvars) );
   ndemands = 0;

   /* get demand array to initialize knapsack problem */
   SCIP_CALL( collectDemands(scip, consdata, startindices, curtime, nstarted, nfinished, &demands, &ndemands) );

   /* create array for profits */
   SCIP_CALL( SCIPallocBufferArray(scip, &profits, ndemands) );
   SCIP_CALL( SCIPallocBufferArray(scip, &items, ndemands) );
   for( j = 0; j < ndemands; ++j )
   {
      profits[j] = (SCIP_Real) demands[j];
      items[j] = j;/* this is only a dummy value*/
   }

   /* solve knapsack problem and get maximum capacity usage <= capacity */
   SCIP_CALL( SCIPsolveKnapsackExactly(scip, ndemands, demands, profits, (SCIP_Longint)consdata->capacity,
         items, NULL, NULL, NULL, NULL, &solval, &success) );

   assert(SCIPisFeasIntegral(scip, solval));

   /* store result */
   *bestcapacity = SCIPconvertRealToInt(scip, solval);

   SCIPfreeBufferArray(scip, &items);
   SCIPfreeBufferArray(scip, &profits);
   SCIPfreeBufferArray(scip, &demands);

   return SCIP_OKAY;
}

/** try to tighten the capacity
 *  -- using DP for knapsack, we find the maximum possible capacity usage
 *  -- neglects hmin and hmax, such that it is also able to check solutions globally
 */
static
SCIP_RETCODE tightenCapacity(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  nchgcoefs,          /**< pointer to count total number of changed coefficients */
   int*                  nchgsides           /**< pointer to store the number of changed sides */
   )
{
   SCIP_CONSDATA* consdata;
   int* starttimes;         /* stores when each job is starting */
   int* endtimes;           /* stores when each job ends */
   int* startindices;       /* we will sort the startsolvalues, thus we need to know wich index of a job it corresponds to */
   int* endindices;         /* we will sort the endsolvalues, thus we need to know wich index of a job it corresponds to */

   int nvars;               /* number of activities for this constraint */
   int freecapacity;        /* remaining capacity */
   int curtime;             /* point in time which we are just checking */
   int endindex;            /* index of endsolvalues with: endsolvalues[endindex] > curtime */

   int bestcapacity;

   int j;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(nchgsides != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   /* if no activities are associated with this cumulative or the capacity is 1, then this constraint is redundant */
   if( nvars <= 1 || consdata->capacity <= 1 )
      return SCIP_OKAY;

   assert(consdata->vars != NULL);

   SCIPdebugMessage("try to tighten capacity for cumulative constraint <%s> with capacity %d\n",
      SCIPconsGetName(cons), consdata->capacity);

   SCIP_CALL( SCIPallocBufferArray(scip, &starttimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endtimes, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &startindices, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &endindices, nvars) );

   /* create event point arrays */
   createSortedEventpoints(scip, nvars, consdata->vars, consdata->durations,
      starttimes, endtimes, startindices, endindices, FALSE);

   bestcapacity = 1;
   endindex = 0;
   freecapacity = consdata->capacity;

   /* check each startpoint of a job whether the capacity is kept or not */
   for( j = 0; j < nvars && bestcapacity < consdata->capacity; ++j )
   {
      curtime = starttimes[j];
      SCIPdebugMessage("look at %d-th job with start %d\n", j, curtime);

      /* remove the capacity requirments for all job which start at the curtime */
      subtractStartingJobDemands(consdata, curtime, starttimes, startindices, &freecapacity, &j, nvars);

      /* add the capacity requirments for all job which end at the curtime */
      addEndingJobDemands(consdata, curtime, endtimes, endindices, &freecapacity, &endindex, nvars);

      assert(freecapacity <= consdata->capacity);
      assert(endindex <= nvars);

      /* endindex - points to the next job which will finish */
      /* j - points to the last job that has been released */

      /* check point in time when capacity is exceeded (here, a knapsack problem must be solved) */
      if( freecapacity < 0 )
      {
         int newcapacity;

         newcapacity = 1;

         /* get best possible upper bound on capacity usage */
         SCIP_CALL( getHighestCapacityUsage(scip, cons, startindices, curtime, j+1, endindex, &newcapacity) );

         /* update bestcapacity */
         bestcapacity = MAX(bestcapacity, newcapacity);
         SCIPdebugMessage("after highest cap usage: bestcapacity = %d\n", bestcapacity);
      }

      /* also those points in time, where the capacity limit is not exceeded, must be taken into account */
      if( freecapacity > 0 && freecapacity != consdata->capacity )
      {
         bestcapacity = MAX(bestcapacity, consdata->capacity - freecapacity);
         SCIPdebugMessage("after peak < cap: bestcapacity = %d\n", bestcapacity);
      }

      /* capacity cannot be decreased if the demand sum over more than one job equals the capacity */
      if( freecapacity == 0 && consdata->demands[startindices[j]] < consdata->capacity)
      {
         /* if demands[startindices[j]] == cap then exactly that job is running */
         SCIPdebugMessage("--> cannot decrease capacity since sum equals capacity\n");
         bestcapacity = consdata->capacity;
         break;
      }
   }  /*lint --e{850}*/

   /* free all buffer arrays */
   SCIPfreeBufferArray(scip, &endindices);
   SCIPfreeBufferArray(scip, &startindices);
   SCIPfreeBufferArray(scip, &endtimes);
   SCIPfreeBufferArray(scip, &starttimes);

   /* check whether capacity can be tightened and whether demands need to be adjusted */
   if( bestcapacity < consdata->capacity )
   {
      int oldnchgcoefs;

      oldnchgcoefs = *nchgcoefs;

      SCIPdebugMessage("+-+-+-+-+-+ --> CHANGE capacity of cons<%s> from %d to %d\n",
         SCIPconsGetName(cons), consdata->capacity, bestcapacity);

      for( j = 0; j < nvars; ++j )
      {
         if( consdata->demands[j] == consdata->capacity )
         {
            consdata->demands[j] = bestcapacity;
            (*nchgcoefs)++;
         }
      }

      consdata->capacity = bestcapacity;
      (*nchgsides)++;

      SCIPdebugPrintf("; changed additionally %d coefficients\n", (*nchgcoefs)-oldnchgcoefs);

      consdata->varbounds = FALSE;
   }

   return SCIP_OKAY;
}

/** tries to change coefficients:
 *        demand_j < cap && all other parallel jobs in conflict
 *         ==> set demand_j := cap
 */
static
SCIP_RETCODE tightenCoefs(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  nchgcoefs           /**< pointer to count total number of changed coefficients */
   )
{
   SCIP_CONSDATA* consdata;
   int nvars;
   int j;
   int oldnchgcoefs;
   int mindemand;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(nchgcoefs != NULL);

   /* get constraint data for some parameter testings only! */
   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;
   oldnchgcoefs = *nchgcoefs;

   if( nvars <= 0 )
      return SCIP_OKAY;

   /* PRE1:
    * check all jobs j whether: r_j + r_min > capacity holds
    * if so: adjust r_j to capacity
    */
   mindemand = consdata->demands[0];
   for( j = 0; j < nvars; ++j )
   {
      mindemand = MIN(mindemand, consdata->demands[j]);
   }

   /*check each job */
   for( j = 0; j < nvars; ++j )
   {
      if( mindemand + consdata->demands[j] > consdata->capacity  && consdata->demands[j] < consdata->capacity )
      {
         SCIPdebugMessage("+-+-+-+-+-+change demand of var<%s> from %d to capacity %d\n", SCIPvarGetName(consdata->vars[j]),
            consdata->demands[j], consdata->capacity);
         consdata->demands[j] = consdata->capacity;
         (*nchgcoefs)++;
      }
   }

   /* PRE2:
    * check for each job (with d_j < cap)
    * whether it is disjunctive to all others over the time horizon
    */
   for( j = 0; j < nvars; ++j )
   {
      SCIP_Bool chgcoef;
      int est_j;
      int lct_j;
      int i;

      assert(consdata->demands[j] <= consdata->capacity);

      if( consdata->demands[j] == consdata->capacity )
         continue;

      chgcoef = TRUE;

      est_j = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[j]));
      lct_j = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[j])) + consdata->durations[j];

      for( i = 0; i < nvars; ++i )
      {
         int est_i;
         int lct_i;

         if( i == j )
            continue;

         est_i = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(consdata->vars[i]));
         lct_i = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(consdata->vars[i])) + consdata->durations[i];

         if( est_i >= lct_j || est_j >= lct_i )
            continue;

         if( consdata->demands[j] + consdata->demands[i] <= consdata->capacity )
         {
            chgcoef = FALSE;
            break;
         }
      }

      if( chgcoef )
      {
         SCIPdebugMessage("+-+-+-+-+-+change demand of var<%s> from %d to capacity %d\n", SCIPvarGetName(consdata->vars[j]),
            consdata->demands[j], consdata->capacity);
         consdata->demands[j] = consdata->capacity;
         (*nchgcoefs)++;
      }

   }

   if( (*nchgcoefs) > oldnchgcoefs )
   {
      SCIPdebugMessage("+-+-+-+-+-+changed %d coefficients of variables of cumulative constraint<%s>\n",
         (*nchgcoefs) - oldnchgcoefs, SCIPconsGetName(cons));
   }

   return SCIP_OKAY;
}

#if 0
/** try to reformulate constraint by replacing certain jobs */
static
SCIP_RETCODE reformulateCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  naggrvars           /**< pointer to store the number of aggregated variables */
   )
{
   SCIP_CONSDATA* consdata;
   int hmin;
   int hmax;
   int nvars;
   int v;

   consdata = SCIPconsGetData(cons);
   assert(cons != NULL);

   nvars = consdata->nvars;
   assert(nvars > 1);

   hmin = consdata->hmin;
   hmax = consdata->hmax;
   assert(hmin < hmax);

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;
      int duration;
      int est;
      int ect;
      int lst;
      int lct;

      var = consdata->vars[v];
      assert(var != NULL);

      duration = consdata->durations[v];

      est = SCIPconvertRealToInt(scip, SCIPvarGetLbGlobal(var));
      ect = est + duration;
      lst = SCIPconvertRealToInt(scip, SCIPvarGetUbGlobal(var));
      lct = lst + duration;

      /* jobs for which the core [lst,ect) contains [hmin,hmax) should be removed already */
      assert(lst > hmin || ect < hmax);

      if( lst <= hmin && est < hmin - lct + MIN(hmin, ect) )
      {
         SCIP_VAR* aggrvar;
         char name[SCIP_MAXSTRLEN];
         SCIP_Bool infeasible;
         SCIP_Bool redundant;
         SCIP_Bool aggregated;
         int shift;

         shift = est - (hmin - lct + MIN(hmin, ect));
         assert(shift > 0);
         lst = hmin;
         duration = hmin - lct;

         SCIPdebugMessage("replace variable <%s>[%g,%g] by [%d,%d]\n",
            SCIPvarGetName(var), SCIPvarGetLbGlobal(var), SCIPvarGetUbGlobal(var), est + shift, lst);

         (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "%s_aggr", SCIPvarGetName(var));
         SCIP_CALL( SCIPcreateVar(scip, &aggrvar, name, (SCIP_Real)(est+shift), (SCIP_Real)lst, 0.0, SCIPvarGetType(var),
               SCIPvarIsInitial(var), SCIPvarIsRemovable(var), NULL, NULL, NULL, NULL, NULL) );
         SCIP_CALL( SCIPaddVar(scip, var) );
         SCIP_CALL( SCIPaggregateVars(scip, var, aggrvar, 1.0, -1.0, (SCIP_Real)shift, &infeasible, &redundant, &aggregated) );

         assert(!infeasible);
         assert(!redundant);
         assert(aggregated);

         /* replace variable */
         consdata->durations[v] = duration;
         consdata->vars[v] = aggrvar;

         /* remove and add locks */
         SCIP_CALL( SCIPunlockVarCons(scip, var, cons, consdata->downlocks[v], consdata->uplocks[v]) );
         SCIP_CALL( SCIPlockVarCons(scip, var, cons, consdata->downlocks[v], consdata->uplocks[v]) );

         SCIP_CALL( SCIPreleaseVar(scip, &aggrvar) );

         (*naggrvars)++;
      }
   }

   return SCIP_OKAY;
}
#endif

/** creare a disjunctive constraint which contains all jobs which cannot run in parallel */
static
SCIP_RETCODE createDisjuctiveCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR** vars;
   int* durations;
   int* demands;
   int capacity;
   int halfcapacity;
   int mindemand;
   int nvars;
   int v;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   capacity = consdata->capacity;

   if( capacity == 1 )
      return SCIP_OKAY;

   SCIP_CALL( SCIPallocBufferArray(scip, &vars, consdata->nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &durations, consdata->nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demands, consdata->nvars) );

   halfcapacity = capacity / 2;
   mindemand = consdata->capacity;
   nvars = 0;

   /* collect all jobs with demand larger than half of the capacity */
   for( v = 0; v < consdata->nvars; ++v )
   {
      if( consdata->demands[v] > halfcapacity )
      {
         vars[nvars] = consdata->vars[v];
         demands[nvars] = 1;
         durations[nvars] = consdata->durations[v];
         nvars++;

         mindemand = MIN(mindemand, consdata->demands[v]);
      }
   }

   if( nvars > 0 )
   {
      /* add all jobs which has a demand smaller than one half of the capacity but together with the smallest collected
       * job is still to large to be scheduled in parallel
       */
      for( v = 0; v < consdata->nvars; ++v )
      {
         if( consdata->demands[v] > halfcapacity )
            continue;

         if( mindemand + consdata->demands[v] > capacity )
         {
            demands[nvars] = 1;
            durations[nvars] = consdata->durations[v];
            vars[nvars] = consdata->vars[v];
            nvars++;

            /* @todo create one cumulative constraint and look for another small demand */
            break;
         }
      }

      /* creates cumulative constraint and adds it to problem */
      SCIP_CALL( createConsCumulative(scip, SCIPconsGetName(cons), nvars, vars, durations, demands, 1, consdata->hmin, consdata->hmax,
            FALSE, FALSE, FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE) );
      (*naddconss)++;
   }

   SCIPfreeBufferArray(scip, &demands);
   SCIPfreeBufferArray(scip, &durations);
   SCIPfreeBufferArray(scip, &vars);

   return SCIP_OKAY;
}

/** presolve given constraint */
static
SCIP_RETCODE presolveCons(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< cumulative constraint */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_PRESOLTIMING     presoltiming,       /**< timing of presolving call */
   int*                  nfixedvars,         /**< pointer to store the number of fixed variables */
#if 0
   int*                  naggrvars,          /**< pointer to counter which is increased by the number of deduced variable aggregations */
#endif
   int*                  nchgbds,            /**< pointer to store the number of changed bounds */
   int*                  ndelconss,          /**< pointer to store the number of deleted constraints */
   int*                  naddconss,          /**< pointer to store the number of added constraints */
   int*                  nchgcoefs,          /**< pointer to store the number of changed coefficients */
   int*                  nchgsides,          /**< pointer to store the number of changed sides */
   SCIP_Bool*            cutoff,             /**< pointer to store if a cutoff was detected */
   SCIP_Bool*            unbounded           /**< pointer to store if the problem is unbounded */
   )
{
   assert(!SCIPconsIsDeleted(cons));

   /* only perform dual reductions on model constraints */
   if( conshdlrdata->dualpresolve && SCIPallowDualReds(scip) )
   {
      /* computes the effective horizon and checks if the constraint can be decomposed */
      SCIP_CALL( computeEffectiveHorizon(scip, cons, ndelconss, naddconss, nchgsides) );

      if( SCIPconsIsDeleted(cons) )
         return SCIP_OKAY;

      /* in case the cumulative constraint is independent of every else, solve the cumulative problem and apply the
       * fixings (dual reductions)
       */
      if( (presoltiming & SCIP_PRESOLTIMING_EXHAUSTIVE) != 0 )
      {
         SCIP_CALL( solveIndependentCons(scip, cons, conshdlrdata->maxnodes, nchgbds, nfixedvars, ndelconss, cutoff, unbounded) );

         if( *cutoff || *unbounded || presoltiming == SCIP_PRESOLTIMING_EXHAUSTIVE )
            return SCIP_OKAY;
      }

      SCIP_CALL( presolveConsEffectiveHorizon(scip, cons, nfixedvars, nchgcoefs, nchgsides, cutoff) );

      if( *cutoff || SCIPconsIsDeleted(cons) )
         return SCIP_OKAY;
   }

   /* remove jobs which have a demand larger than the capacity */
   SCIP_CALL( removeOversizedJobs(scip, cons, nchgbds, nchgcoefs, naddconss, cutoff) );
   assert((*cutoff) || checkDemands(scip, cons));

   if( *cutoff )
      return SCIP_OKAY;

   if( conshdlrdata->normalize )
   {
      /* divide demands by their greatest common divisor */
      SCIP_CALL( normalizeDemands(scip, cons, nchgcoefs, nchgsides) );
   }

   /* delete constraint with one job */
   SCIP_CALL( deleteTrivilCons(scip, cons, ndelconss, cutoff) );

   if( *cutoff || SCIPconsIsDeleted(cons) )
      return SCIP_OKAY;

   if( conshdlrdata->coeftightening )
   {
      /* try to tighten the capacity */
      SCIP_CALL( tightenCapacity(scip, cons, nchgcoefs, nchgsides) );

      /* try to tighten the coefficients */
      SCIP_CALL( tightenCoefs(scip, cons, nchgcoefs) );
   }

   assert(checkDemands(scip, cons) || *cutoff);

#if 0
   SCIP_CALL( reformulateCons(scip, cons, naggrvars) );
#endif

   return SCIP_OKAY;
}

/**@name TClique Graph callbacks
 *
 * @{
 */

/** tclique graph data */
struct TCLIQUE_Graph
{
   SCIP_VAR**            vars;               /**< start time variables each of them is a node */
   SCIP_HASHMAP*         varmap;             /**< variable map, mapping variable to indux in vars array */
   SCIP_Bool**           precedencematrix;   /**< precedence adjacent matrix */
   SCIP_Bool**           demandmatrix;       /**< demand adjacent matrix */
   TCLIQUE_WEIGHT*       weights;            /**< weight of nodes */
   int*                  ninarcs;            /**< number if in arcs for the precedence graph */
   int*                  noutarcs;           /**< number if out arcs for the precedence graph */
   int*                  durations;          /**< for each node the duration of the corresponding job */
   int                   nnodes;             /**< number of nodes */
   int                   size;               /**< size of the array */
};

/** gets number of nodes in the graph */
static
TCLIQUE_GETNNODES(tcliqueGetnnodesClique)
{
   assert(tcliquegraph != NULL);

   return tcliquegraph->nnodes;
}

/** gets weight of nodes in the graph */
static
TCLIQUE_GETWEIGHTS(tcliqueGetweightsClique)
{
   assert(tcliquegraph != NULL);

   return tcliquegraph->weights;
}

/** returns, whether the edge (node1, node2) is in the graph */
static
TCLIQUE_ISEDGE(tcliqueIsedgeClique)
{
   assert(tcliquegraph != NULL);
   assert(0 <= node1 && node1 < tcliquegraph->nnodes);
   assert(0 <= node2 && node2 < tcliquegraph->nnodes);

   /* check if an arc exits in the precedence graph */
   if( tcliquegraph->precedencematrix[node1][node2] || tcliquegraph->precedencematrix[node2][node1] )
      return TRUE;

   /* check if an edge exits in the non-overlapping graph */
   if( tcliquegraph->demandmatrix[node1][node2] )
      return TRUE;

   return FALSE;
}

/** selects all nodes from a given set of nodes which are adjacent to a given node
 *  and returns the number of selected nodes
 */
static
TCLIQUE_SELECTADJNODES(tcliqueSelectadjnodesClique)
{
   int nadjnodes;
   int i;

   assert(tcliquegraph != NULL);
   assert(0 <= node && node < tcliquegraph->nnodes);
   assert(nnodes == 0 || nodes != NULL);
   assert(adjnodes != NULL);

   nadjnodes = 0;

   for( i = 0; i < nnodes; i++ )
   {
      /* check if the node is adjacent to the given node (nodes and adjacent nodes are ordered by node index) */
      assert(0 <= nodes[i] && nodes[i] < tcliquegraph->nnodes);
      assert(i == 0 || nodes[i-1] < nodes[i]);

      /* check if an edge exists */
      if( tcliqueIsedgeClique(tcliquegraph, node, nodes[i]) )
      {
         /* current node is adjacent to given node */
         adjnodes[nadjnodes] = nodes[i];
         nadjnodes++;
      }
   }

   return nadjnodes;
}

/** generates cuts using a clique found by algorithm for maximum weight clique
 *  and decides whether to stop generating cliques with the algorithm for maximum weight clique
 */
static
TCLIQUE_NEWSOL(tcliqueNewsolClique)
{  /*lint --e{715}*/
   SCIPdebugMessage("####### max clique %d\n", cliqueweight);
}

/** print the tclique graph */
#if 0
static
void tcliquePrint(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph        /**< tclique graph */
   )
{
   int nnodes;
   int i;
   int j;

   nnodes = tcliquegraph->nnodes;

   for( i = 0; i < nnodes; ++i )
   {
      for( j = 0; j < nnodes; ++j )
      {
         SCIPinfoMessage(scip, NULL, "(%d/%d) ", tcliquegraph->precedencematrix[i][j], tcliquegraph->demandmatrix[i][j]);
      }
      SCIPinfoMessage(scip, NULL, "\n");
   }
}
#endif

/** @} */

/** analyzes if the given variable lower bound condition implies a precedence condition w.r.t. given duration for the
 *  job corresponding to variable bound variable (vlbvar)
 *
 *  variable lower bound is given as:  var >= vlbcoef * vlbvar + vlbconst
 */
static
SCIP_Bool impliesVlbPrecedenceCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             vlbvar,             /**< variable which bounds the variable from below */
   SCIP_Real             vlbcoef,            /**< variable bound coefficient */
   SCIP_Real             vlbconst,           /**< variable bound constant */
   int                   duration            /**< duration of the variable bound variable */
   )
{
   if( SCIPisEQ(scip, vlbcoef, 1.0) )
   {
      if( SCIPisGE(scip, vlbconst, (SCIP_Real) duration) )
      {
         /* if vlbcoef = 1 and vlbcoef >= duration -> precedence condition */
         return TRUE;
      }
   }
   else
   {
      SCIP_Real bound;

      bound = (duration -  vlbcoef) / (vlbcoef - 1.0);

      if( SCIPisLT(scip, vlbcoef, 1.0) )
      {
         SCIP_Real ub;

         ub = SCIPvarGetUbLocal(vlbvar);

         /* if vlbcoef < 1 and ub(vlbvar) <= (duration - vlbconst)/(vlbcoef - 1) -> precedence condition */
         if( SCIPisLE(scip, ub, bound) )
            return TRUE;
      }
      else
      {
         SCIP_Real lb;

         assert(SCIPisGT(scip, vlbcoef, 1.0));

         lb = SCIPvarGetLbLocal(vlbvar);

         /* if  vlbcoef > 1 and lb(vlbvar) >= (duration - vlbconst)/(vlbcoef - 1) -> precedence condition */
         if( SCIPisGE(scip, lb, bound) )
            return TRUE;
      }
   }

   return FALSE;
}

/** analyzes if the given variable upper bound condition implies a precedence condition w.r.t. given duration for the
 *  job corresponding to variable which is bounded (var)
 *
 *  variable upper bound is given as:  var <= vubcoef * vubvar + vubconst
 */
static
SCIP_Bool impliesVubPrecedenceCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_VAR*             var,                /**< variable which is bound from above */
   SCIP_Real             vubcoef,            /**< variable bound coefficient */
   SCIP_Real             vubconst,           /**< variable bound constant */
   int                   duration            /**< duration of the variable which is bounded from above */
   )
{
   SCIP_Real vlbcoef;
   SCIP_Real vlbconst;

   /* convert the variable upper bound into an variable lower bound */
   vlbcoef = 1.0 / vubcoef;
   vlbconst = -vubconst / vubcoef;

   return impliesVlbPrecedenceCondition(scip, var, vlbcoef, vlbconst, duration);
}

/** get the corresponding index of the given variables; this in case of an active variable the problem index and for
 *  others an index larger than the number if active variables
 */
static
SCIP_RETCODE getNodeIdx(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< incompatibility graph */
   SCIP_VAR*             var,                /**< variable for which we want the index */
   int*                  idx                 /**< pointer to store the index */
   )
{
   (*idx) = SCIPvarGetProbindex(var);

   if( (*idx) == -1 )
   {
      if( SCIPhashmapExists(tcliquegraph->varmap, (void*)var) )
      {
         (*idx) = (int)(size_t) SCIPhashmapGetImage(tcliquegraph->varmap, (void*)var);
      }
      else
      {
         int pos;
         int v;

         /**@todo we might want to add the aggregation path to graph */

         /* check if we have to realloc memory */
         if( tcliquegraph->size == tcliquegraph->nnodes )
         {
            int size;

            size = SCIPcalcMemGrowSize(scip, tcliquegraph->nnodes+1);
            tcliquegraph->size = size;

            SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->vars, size) );
            SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->precedencematrix, size) );
            SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->demandmatrix, size) );
            SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->durations, size) );
            SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->weights, size) );

            for( v = 0; v < tcliquegraph->nnodes; ++v )
            {
               SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->precedencematrix[v], size) ); /*lint !e866*/
               SCIP_CALL( SCIPreallocBufferArray(scip, &tcliquegraph->demandmatrix[v], size) ); /*lint !e866*/
            }
         }
         assert(tcliquegraph->nnodes < tcliquegraph->size);

         pos = tcliquegraph->nnodes;
         assert(pos >= 0);

         tcliquegraph->durations[pos] = 0;
         tcliquegraph->weights[pos] = 0;
         tcliquegraph->vars[pos] = var;

         SCIP_CALL( SCIPallocBufferArray(scip, &tcliquegraph->precedencematrix[pos], tcliquegraph->size) ); /*lint !e866*/
         BMSclearMemoryArray(tcliquegraph->precedencematrix[pos], tcliquegraph->nnodes); /*lint !e866*/

         SCIP_CALL( SCIPallocBufferArray(scip, &tcliquegraph->demandmatrix[pos], tcliquegraph->size) ); /*lint !e866*/
         BMSclearMemoryArray(tcliquegraph->demandmatrix[pos], tcliquegraph->nnodes); /*lint !e866*/

         SCIP_CALL( SCIPhashmapInsert(tcliquegraph->varmap, (void*)var, (void*)(size_t)(pos)) );

         tcliquegraph->nnodes++;

         for( v = 0; v < tcliquegraph->nnodes; ++v )
         {
            tcliquegraph->precedencematrix[v][pos] = 0;
            tcliquegraph->demandmatrix[v][pos] = 0;
         }

         (*idx) = tcliquegraph->nnodes;
      }
   }
   else
   {
      assert(*idx == (int)(size_t)SCIPhashmapGetImage(tcliquegraph->varmap, (void*)var));
   }

   assert(SCIPhashmapExists(tcliquegraph->varmap, (void*)var));

   return SCIP_OKAY;
}

/** use the variables bounds of SCIP to projected variables bound graph into a precedence garph
 *
 *  Let d be the (assumed) duration of variable x and consider a variable bound of the form b * x + c <= y. This
 *  variable bounds implies a precedence condition x -> y (meaning job y starts after job x is finished) if:
 *
 *  (i)   b = 1 and c  >= d
 *  (ii)  b > 1 and lb(x) >= (d - c)/(b - 1)
 *  (iii) b < 1 and ub(x) >= (d - c)/(b - 1)
 *
 */
static
SCIP_RETCODE projectVbd(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph        /**< incompatibility graph */
   )
{
   SCIP_VAR** vars;
   int nvars;
   int v;

   vars = SCIPgetVars(scip);
   nvars = SCIPgetNVars(scip);

   /* try to project each arc of the variable bound graph to precedence condition */
   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR** vbdvars;
      SCIP_VAR* var;
      SCIP_Real* vbdcoefs;
      SCIP_Real* vbdconsts;
      int nvbdvars;
      int idx1;
      int b;

      var = vars[v];
      assert(var != NULL);

      SCIP_CALL( getNodeIdx(scip, tcliquegraph, var, &idx1) );
      assert(idx1 >= 0);

      if( tcliquegraph->durations[idx1] == 0 )
         continue;

      vbdvars = SCIPvarGetVlbVars(var);
      vbdcoefs = SCIPvarGetVlbCoefs(var);
      vbdconsts = SCIPvarGetVlbConstants(var);
      nvbdvars = SCIPvarGetNVlbs(var);

      for( b = 0; b < nvbdvars; ++b )
      {
         int idx2;

         SCIP_CALL( getNodeIdx(scip, tcliquegraph, vbdvars[b], &idx2) );
         assert(idx2 >= 0);

         if( tcliquegraph->durations[idx2] == 0 )
            continue;

         if( impliesVlbPrecedenceCondition(scip, vbdvars[b], vbdcoefs[b], vbdconsts[b], tcliquegraph->durations[idx2]) )
            tcliquegraph->precedencematrix[idx2][idx1] = TRUE;
      }

      vbdvars = SCIPvarGetVubVars(var);
      vbdcoefs = SCIPvarGetVubCoefs(var);
      vbdconsts = SCIPvarGetVubConstants(var);
      nvbdvars = SCIPvarGetNVubs(var);

      for( b = 0; b < nvbdvars; ++b )
      {
         int idx2;

         SCIP_CALL( getNodeIdx(scip, tcliquegraph, vbdvars[b], &idx2) );
         assert(idx2 >= 0);

         if( tcliquegraph->durations[idx2] == 0 )
            continue;

         if( impliesVubPrecedenceCondition(scip, var, vbdcoefs[b], vbdconsts[b], tcliquegraph->durations[idx1]) )
            tcliquegraph->precedencematrix[idx1][idx2] = TRUE;
      }

      for( b = v+1; b < nvars; ++b )
      {
         int idx2;

         SCIP_CALL( getNodeIdx(scip, tcliquegraph, vars[b], &idx2) );
         assert(idx2 >= 0);

         if( tcliquegraph->durations[idx2] == 0 )
            continue;

         /* check if the latest completion time of job1 is smaller than the earliest start time of job2 */
         if( SCIPisLE(scip, SCIPvarGetUbLocal(var) + tcliquegraph->durations[idx1], SCIPvarGetLbLocal(vars[b])) )
            tcliquegraph->precedencematrix[idx1][idx2] = TRUE;

         /* check if the latest completion time of job2 is smaller than the earliest start time of job1 */
         if( SCIPisLE(scip, SCIPvarGetUbLocal(vars[b]) + tcliquegraph->durations[idx2], SCIPvarGetLbLocal(var)) )
            tcliquegraph->precedencematrix[idx2][idx1] = TRUE;
      }
   }

   return SCIP_OKAY;
}

/** compute the transitive closer of the given graph and the number of in and out arcs */
static
void transitiveClosure(
   SCIP_Bool**           adjmatrix,          /**< adjacent matrix */
   int*                  ninarcs,            /**< array to store the number of in arcs */
   int*                  noutarcs,           /**< array to store the number of out arcs */
   int                   nnodes              /**< number if nodes */
   )
{
   int i;
   int j;
   int k;

   for( i = 0; i < nnodes; ++i )
   {
      for( j = 0; j < nnodes; ++j )
      {
         if( adjmatrix[i][j] )
         {
            ninarcs[j]++;
            noutarcs[i]++;

            for( k = 0; k < nnodes; ++k )
            {
               if( adjmatrix[j][k] )
                  adjmatrix[i][k] = TRUE;
            }
         }
      }
   }
}

/** constructs a non-overlapping graph w.r.t. given durations and available cumulative constraints */
static
SCIP_RETCODE constraintNonOverlappingGraph(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< incompatibility graph */
   SCIP_CONS**           conss,              /**< array of cumulative constraints */
   int                   nconss              /**< number of cumulative constraints */
   )
{
   int c;

   /* use the cumulative constraints to initialize the none overlapping graph */
   for( c = 0; c < nconss; ++c )
   {
      SCIP_CONSDATA* consdata;
      SCIP_VAR** vars;
      int* demands;
      int capacity;
      int nvars;
      int i;

      consdata = SCIPconsGetData(conss[c]);
      assert(consdata != NULL);

      vars = consdata->vars;
      demands = consdata->demands;

      nvars = consdata->nvars;
      capacity = consdata->capacity;

      SCIPdebugMessage("constraint <%s>\n", SCIPconsGetName(conss[c]));

      /* check pairwise if two jobs have a cumulative demand larger than the capacity */
      for( i = 0; i < nvars; ++i )
      {
         int idx1;
         int j;

         SCIP_CALL( getNodeIdx(scip, tcliquegraph, vars[i], &idx1) );
         assert(idx1 >= 0);

         if( tcliquegraph->durations[idx1] == 0 || tcliquegraph->durations[idx1] > consdata->durations[i] )
            continue;

         for( j = i+1; j < nvars; ++j )
         {
            assert(consdata->durations[j] > 0);

            if( demands[i] + demands[j] > capacity )
            {
               int idx2;
               int est1;
               int est2;
               int lct1;
               int lct2;

               /* check if the effective horizon is large enough */
               est1 = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[i]));
               est2 = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[j]));

               /* at least one of the jobs needs to start at hmin or later */
               if( est1 < consdata->hmin && est2 < consdata->hmin )
                  continue;

               lct1 = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[i])) + consdata->durations[i];
               lct2 = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[j])) + consdata->durations[j];

               /* at least one of the jobs needs to finish not later then hmin */
               if( lct1 > consdata->hmax && lct2 > consdata->hmax )
                  continue;

               SCIP_CALL( getNodeIdx(scip, tcliquegraph, vars[j], &idx2) );
               assert(idx2 >= 0);
               assert(idx1 != idx2);

               if( tcliquegraph->durations[idx2] == 0 || tcliquegraph->durations[idx2] > consdata->durations[j] )
                  continue;

               SCIPdebugMessage(" *** variable <%s> and variable <%s>\n", SCIPvarGetName(vars[i]), SCIPvarGetName(vars[j]));

               assert(tcliquegraph->durations[idx1] > 0);
               assert(tcliquegraph->durations[idx2] > 0);

               tcliquegraph->demandmatrix[idx1][idx2] = TRUE;
               tcliquegraph->demandmatrix[idx2][idx1] = TRUE;

            }
         }
      }
   }

   return SCIP_OKAY;
}

/** constructs a conflict set graph (undirected) which contains for each job a node and edge if the corresponding pair
 *  of jobs cannot run in parallel
 */
static
SCIP_RETCODE constructIncompatibilityGraph(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< incompatibility graph */
   SCIP_CONS**           conss,              /**< array of cumulative constraints */
   int                   nconss              /**< number of cumulative constraints */
   )
{
   assert(scip != NULL);
   assert(tcliquegraph != NULL);

   /* use the variables bounds of SCIP to project the variables bound graph inot a precedence graph */
   SCIP_CALL( projectVbd(scip, tcliquegraph) );

   /* compute the transitive closure of the precedence graph and the number of in and out arcs */
   transitiveClosure(tcliquegraph->precedencematrix, tcliquegraph->ninarcs, tcliquegraph->noutarcs, tcliquegraph->nnodes);

   /* constraints non-overlapping graph */
   SCIP_CALL( constraintNonOverlappingGraph(scip, tcliquegraph, conss, nconss) );

   return SCIP_OKAY;
}

/** create cumulative constraint from conflict set */
static
SCIP_RETCODE createCumulativeCons(
   SCIP*                 scip,               /**< SCIP data structure */
   const char*           name,               /**< constraint name */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< conflict set graph */
   int*                  cliquenodes,        /**< array storing the indecies of the nodes belonging to the clique */
   int                   ncliquenodes        /**< number of nodes in the clique */
   )
{
   SCIP_CONS* cons;
   SCIP_VAR** vars;
   int* durations;
   int* demands;
   int v;

   SCIP_CALL( SCIPallocBufferArray(scip, &vars, ncliquenodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &durations, ncliquenodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demands, ncliquenodes) );

   SCIPsortInt(cliquenodes, ncliquenodes);

   /* collect variables, durations, and demands */
   for( v = 0; v < ncliquenodes; ++v )
   {
      durations[v] = tcliquegraph->durations[cliquenodes[v]];
      assert(durations[v] > 0);
      demands[v] = 1;
      vars[v] = tcliquegraph->vars[cliquenodes[v]];
   }

   /* create (unary) cumulative constraint */
   SCIP_CALL( SCIPcreateConsCumulative(scip, &cons, name, ncliquenodes, vars, durations, demands, 1,
         FALSE, TRUE, TRUE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE) );

   SCIP_CALL( SCIPaddCons(scip, cons) );
   SCIP_CALL( SCIPreleaseCons(scip, &cons) );

   /* free buffers */
   SCIPfreeBufferArray(scip, &demands);
   SCIPfreeBufferArray(scip, &durations);
   SCIPfreeBufferArray(scip, &vars);

   return SCIP_OKAY;
}

/** search for cumulative constrainst */
static
SCIP_RETCODE findCumulativeConss(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< conflict set graph */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   TCLIQUE_STATUS tcliquestatus;
   SCIP_Bool* precedencerow;
   SCIP_Bool* precedencecol;
   SCIP_Bool* demandrow;
   SCIP_Bool* demandcol;
   SCIP_HASHTABLE* covered;
   int* cliquenodes;
   int ncliquenodes;
   int cliqueweight;
   int ntreenodes;
   int nnodes;
   int nconss;
   int v;

   nnodes = tcliquegraph->nnodes;
   nconss = 0;

   /* initialize the weight of each job with its duration */
   for( v = 0; v < nnodes; ++v )
   {
      tcliquegraph->weights[v] = tcliquegraph->durations[v];
   }

   SCIP_CALL( SCIPallocBufferArray(scip, &cliquenodes, nnodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &precedencerow, nnodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &precedencecol, nnodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demandrow, nnodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demandcol, nnodes) );

   /* create a hash table to store all start time variables which are already covered by at least one clique */
   SCIP_CALL( SCIPhashtableCreate(&covered, SCIPblkmem(scip), SCIPcalcHashtableSize(nnodes),
         SCIPvarGetHashkey, SCIPvarIsHashkeyEq, SCIPvarGetHashkeyVal, NULL) );

   /* for each variables/job we are ... */
   for( v = 0; v < nnodes && !SCIPisStopped(scip); ++v )
   {
      char name[SCIP_MAXSTRLEN];
      int c;

      /* jobs with zero durations are skipped */
      if( tcliquegraph->durations[v] == 0 )
         continue;

      /* check if the start time variable is already covered by at least one clique */
      if( SCIPhashtableExists(covered, tcliquegraph->vars[v]) )
         continue;

      SCIPdebugMessage("********** variable <%s>\n", SCIPvarGetName(tcliquegraph->vars[v]));

      /* temporarily remove the connection via the precedence graph */
      for( c = 0; c < nnodes; ++c )
      {
         precedencerow[c] = tcliquegraph->precedencematrix[v][c];
         precedencecol[c] = tcliquegraph->precedencematrix[c][v];

         demandrow[c] = tcliquegraph->demandmatrix[v][c];
         demandcol[c] = tcliquegraph->demandmatrix[c][v];

#if 0
         if( precedencerow[c] || precedencecol[c] )
         {
            tcliquegraph->demandmatrix[v][c] = FALSE;
            tcliquegraph->demandmatrix[c][v] = FALSE;
         }
#endif

         tcliquegraph->precedencematrix[c][v] = FALSE;
         tcliquegraph->precedencematrix[v][c] = FALSE;
      }

      /* find (heuristically) maximum cliques which includes node v */
      tcliqueMaxClique(tcliqueGetnnodesClique, tcliqueGetweightsClique, tcliqueIsedgeClique, tcliqueSelectadjnodesClique,
         tcliquegraph, tcliqueNewsolClique, NULL,
         cliquenodes, &ncliquenodes, &cliqueweight, 1, 1,
         10000, 1000, 1000, v, &ntreenodes, &tcliquestatus);

      SCIPdebugMessage("tree nodes %d clique size %d (weight %d, status %d)\n", ntreenodes, ncliquenodes, cliqueweight, tcliquestatus);

      if( ncliquenodes == 1 )
         continue;

      /* construct constraint name */
      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "nooverlap_%d_%d", SCIPgetNRuns(scip), nconss);

      SCIP_CALL( createCumulativeCons(scip, name, tcliquegraph, cliquenodes, ncliquenodes) );
      nconss++;

      /* all start time variable to covered hash table */
      for( c = 0; c < ncliquenodes; ++c )
      {
         SCIP_CALL( SCIPhashtableInsert(covered, tcliquegraph->vars[cliquenodes[c]]) );
      }

      /* copy the precedence relations back */
      for( c = 0; c < nnodes; ++c )
      {
         tcliquegraph->precedencematrix[v][c] = precedencerow[c];
         tcliquegraph->precedencematrix[c][v] = precedencecol[c];

         tcliquegraph->demandmatrix[v][c] = demandrow[c];
         tcliquegraph->demandmatrix[c][v] = demandcol[c];
      }
   }

   SCIPhashtableFree(&covered);

   SCIPfreeBufferArray(scip, &demandcol);
   SCIPfreeBufferArray(scip, &demandrow);
   SCIPfreeBufferArray(scip, &precedencecol);
   SCIPfreeBufferArray(scip, &precedencerow);
   SCIPfreeBufferArray(scip, &cliquenodes);

   (*naddconss) += nconss;

   /* for the statistic we count the number added disjunctive constraints */
   SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->naddeddisjunctives += nconss );

   return SCIP_OKAY;
}

/** create precedence constraint (as variable bound constraint */
static
SCIP_RETCODE createPrecedenceCons(
   SCIP*                 scip,               /**< SCIP data structure */
   const char*           name,               /**< constraint name */
   SCIP_VAR*             var,                /**< variable x that has variable bound */
   SCIP_VAR*             vbdvar,             /**< binary, integer or implicit integer bounding variable y */
   int                   distance            /**< minimum distance between the start time of the job corresponding to var and the job corresponding to vbdvar */
   )
{
   SCIP_CONS* cons;

   /* create variable bound constraint */
   SCIP_CALL( SCIPcreateConsVarbound(scip, &cons, name, var, vbdvar, -1.0, -SCIPinfinity(scip), -(SCIP_Real)distance,
         TRUE, TRUE, TRUE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE) );

   SCIPdebugPrintCons(scip, cons, NULL);

   /* add constraint to problem and release it */
   SCIP_CALL( SCIPaddCons(scip, cons) );
   SCIP_CALL( SCIPreleaseCons(scip, &cons) );

   return SCIP_OKAY;
}

/** compute a minimum distance between the start times of the two given jobs and post it as variable bound constraint */
static
SCIP_RETCODE computeMinDistance(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< conflict set graph */
   int                   source,             /**< index of the source node */
   int                   sink,               /**< index of the sink node */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   TCLIQUE_WEIGHT cliqueweight;
   TCLIQUE_STATUS tcliquestatus;
   SCIP_VAR** vars;
   int* cliquenodes;
   int nnodes;
   int lct;
   int est;
   int i;

   int ntreenodes;
   int ncliquenodes;

   /* check if source and sink are connencted */
   if( !tcliquegraph->precedencematrix[source][sink] )
      return SCIP_OKAY;

   nnodes = tcliquegraph->nnodes;
   vars = tcliquegraph->vars;

   /* reset the weights to zero */
   BMSclearMemoryArray(tcliquegraph->weights, nnodes);

   /* get latest completion time (lct) of the source and the earliest start time (est) of sink */
   lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(vars[source])) + tcliquegraph->durations[source];
   est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(vars[sink]));

   /* weight all jobs which run for sure between source and sink with their duration */
   for( i = 0; i < nnodes; ++i )
   {
      SCIP_VAR* var;
      int duration;

      var = vars[i];
      assert(var != NULL);

      duration = tcliquegraph->durations[i];

      if( i == source || i == sink )
      {
         /* source and sink are not weighted */
         tcliquegraph->weights[i] = 0;
      }
      else if( tcliquegraph->precedencematrix[source][i] && tcliquegraph->precedencematrix[i][sink] )
      {
         /* job i runs after source and before sink */
         tcliquegraph->weights[i] = duration;
      }
      else if( lct <= SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var))
         && est >= SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration )
      {
         /* job i run in between due the bounds of the start time variables */
         tcliquegraph->weights[i] = duration;
      }
      else
         tcliquegraph->weights[i] = 0;
   }

   SCIP_CALL( SCIPallocBufferArray(scip, &cliquenodes, nnodes) );

   /* find (heuristically) maximum cliques */
   tcliqueMaxClique(tcliqueGetnnodesClique, tcliqueGetweightsClique, tcliqueIsedgeClique, tcliqueSelectadjnodesClique,
      tcliquegraph, tcliqueNewsolClique, NULL,
      cliquenodes, &ncliquenodes, &cliqueweight, 1, 1,
      10000, 1000, 1000, -1, &ntreenodes, &tcliquestatus);

   if( ncliquenodes > 1 )
   {
      char name[SCIP_MAXSTRLEN];
      int distance;

      /* construct constraint name */
      (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "varbound_%d_%d", SCIPgetNRuns(scip), *naddconss);

      /* the minimum distance between the start times of source job and the sink job is the clique weight plus the
       * duration of the source job
       */
      distance = cliqueweight + tcliquegraph->durations[source];

      SCIP_CALL( createPrecedenceCons(scip, name, vars[source], vars[sink], distance) );
      (*naddconss)++;
   }

   SCIPfreeBufferArray(scip, &cliquenodes);

   return SCIP_OKAY;
}

/** search for precedence constraints
 *
 *  for each arc of the transitive closure of the precedence graph, we are computing a minimum distance between the
 *  corresponding two jobs
 */
static
SCIP_RETCODE findPrecedenceConss(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< conflict set graph */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   int* sources;
   int* sinks;
   int nconss;
   int nnodes;
   int nsources;
   int nsinks;
   int i;

   nnodes = tcliquegraph->nnodes;
   nconss = 0;

   nsources = 0;
   nsinks = 0;

   SCIP_CALL( SCIPallocBufferArray(scip, &sources, nnodes) );
   SCIP_CALL( SCIPallocBufferArray(scip, &sinks, nnodes) );

   /* first collect all sources and sinks */
   for( i = 0; i < nnodes; ++i )
   {
      if( tcliquegraph->ninarcs[i] == 0 )
       {
          sources[nsources] = i;
          nsources++;
       }

      if( tcliquegraph->ninarcs[i] == 0 )
       {
          sinks[nsinks] = i;
          nsinks++;
       }
   }

   /* compute for each node a minimum distance to each sources and each sink */
   for( i = 0; i < nnodes && !SCIPisStopped(scip); ++i )
   {
      int j;

      for( j = 0; j < nsources && !SCIPisStopped(scip); ++j )
      {
         SCIP_CALL( computeMinDistance(scip, tcliquegraph, sources[j], i, &nconss) );
      }

      for( j = 0; j < nsinks && !SCIPisStopped(scip); ++j )
      {
         SCIP_CALL( computeMinDistance(scip, tcliquegraph, i, sinks[j], &nconss) );
      }
   }

   (*naddconss) += nconss;

   /* for the statistic we count the number added variable constraints */
   SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->naddedvarbounds += nconss );

   SCIPfreeBufferArray(scip, &sinks);
   SCIPfreeBufferArray(scip, &sources);

   return SCIP_OKAY;
}

/** initialize the assumed durations for each variable */
static
SCIP_RETCODE initializeDurations(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH*        tcliquegraph,       /**< the incompatibility graph */
   SCIP_CONS**           conss,              /**< cumulative constraints */
   int                   nconss              /**< number of cumulative constraints */
   )
{
   int c;

   /* use the cumulative structure to define the duration we are using for each job */
   for( c = 0; c < nconss; ++c )
   {
      SCIP_CONSDATA* consdata;
      SCIP_VAR** vars;
      int nvars;
      int v;

      consdata = SCIPconsGetData(conss[c]);
      assert(consdata != NULL);

      vars = consdata->vars;
      nvars = consdata->nvars;

      for( v = 0; v < nvars; ++v )
      {
         int idx;

         SCIP_CALL( getNodeIdx(scip, tcliquegraph, vars[v], &idx) );
         assert(idx >= 0);

         /**@todo For the test sets, which we are considere, the durations are independent of the cumulative
          *       constaints. Meaning each job has a fixed duration which is the same for all cumulative constraints. In
          *       general this is not the case. Therefore, the question would be which duration should be used?
          */
         tcliquegraph->durations[idx] = MAX(tcliquegraph->durations[idx], consdata->durations[v]);
         assert(tcliquegraph->durations[idx] > 0);
      }
   }

   return SCIP_OKAY;
}

/** create tclique graph */
static
SCIP_RETCODE createTcliqueGraph(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH**       tcliquegraph        /**< reference to the incompatibility graph */
   )
{
   SCIP_VAR** vars;
   SCIP_HASHMAP* varmap;
   SCIP_Bool** precedencematrix;
   SCIP_Bool** demandmatrix;
   int* ninarcs;
   int* noutarcs;
   int* durations;
   int* weights;
   int nvars;
   int v;

   vars = SCIPgetVars(scip);
   nvars = SCIPgetNVars(scip);

   /* allocate memory for the tclique graph data structure */
   SCIP_CALL( SCIPallocBuffer(scip, tcliquegraph) );

   /* create the variable mapping hash map */
   SCIP_CALL( SCIPhashmapCreate(&varmap, SCIPblkmem(scip), SCIPcalcHashtableSize(5 * nvars)) );

   /* each active variables get a node in the graph */
   SCIP_CALL( SCIPduplicateBufferArray(scip, &(*tcliquegraph)->vars, vars, nvars) );

   /* allocate memory for the projected variables bound graph and the none overlapping graph */
   SCIP_CALL( SCIPallocBufferArray(scip, &precedencematrix, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demandmatrix, nvars) );

   /* array to buffer the weights of the nodes for the maximum weighted clique computation */
   SCIP_CALL( SCIPallocBufferArray(scip, &weights, nvars) );
   BMSclearMemoryArray(weights, nvars);

   /* array to store the number of in arc of the precedence graph */
   SCIP_CALL( SCIPallocBufferArray(scip, &ninarcs, nvars) );
   BMSclearMemoryArray(ninarcs, nvars);

   /* array to store the number of out arc of the precedence graph */
   SCIP_CALL( SCIPallocBufferArray(scip, &noutarcs, nvars) );
   BMSclearMemoryArray(noutarcs, nvars);

   /* array to store the used duration for each node */
   SCIP_CALL( SCIPallocBufferArray(scip, &durations, nvars) );
   BMSclearMemoryArray(durations, nvars);

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR* var;

      var = vars[v];
      assert(var != NULL);

      SCIP_CALL( SCIPallocBufferArray(scip, &precedencematrix[v], nvars) ); /*lint !e866*/
      BMSclearMemoryArray(precedencematrix[v], nvars); /*lint !e866*/

      SCIP_CALL( SCIPallocBufferArray(scip, &demandmatrix[v], nvars) ); /*lint !e866*/
      BMSclearMemoryArray(demandmatrix[v], nvars); /*lint !e866*/

      /* insert all active variables into the garph */
      assert(SCIPvarGetProbindex(var) == v);
      SCIP_CALL( SCIPhashmapInsert(varmap, (void*)var, (void*)(size_t)v) ); /*lint !e571*/
   }

   (*tcliquegraph)->nnodes = nvars;
   (*tcliquegraph)->varmap = varmap;
   (*tcliquegraph)->precedencematrix = precedencematrix;
   (*tcliquegraph)->demandmatrix = demandmatrix;
   (*tcliquegraph)->weights = weights;
   (*tcliquegraph)->ninarcs = ninarcs;
   (*tcliquegraph)->noutarcs = noutarcs;
   (*tcliquegraph)->durations = durations;
   (*tcliquegraph)->size = nvars;

   return SCIP_OKAY;
}

/** frees the tclique graph */
static
void freeTcliqueGraph(
   SCIP*                 scip,               /**< SCIP data structure */
   TCLIQUE_GRAPH**       tcliquegraph        /**< reference to the incompatibility graph */
   )
{
   int v;

   for( v = (*tcliquegraph)->nnodes-1; v >= 0; --v )
   {
      SCIPfreeBufferArray(scip, &(*tcliquegraph)->demandmatrix[v]);
      SCIPfreeBufferArray(scip, &(*tcliquegraph)->precedencematrix[v]);
   }

   SCIPfreeBufferArray(scip, &(*tcliquegraph)->durations);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->ninarcs);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->noutarcs);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->weights);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->demandmatrix);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->precedencematrix);
   SCIPfreeBufferArray(scip, &(*tcliquegraph)->vars);
   SCIPhashmapFree(&(*tcliquegraph)->varmap);

   SCIPfreeBuffer(scip, tcliquegraph);
}

/** construct an incompatibility graph and search for precedence constraints (variables bounds) and unary cumulative
 *  constrains (disjunctive constraint)
 */
static
SCIP_RETCODE detectRedundantConss(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONSHDLRDATA*    conshdlrdata,       /**< constraint handler data */
   SCIP_CONS**           conss,              /**< array of cumulative constraints */
   int                   nconss,             /**< number of cumulative constraints */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   TCLIQUE_GRAPH* tcliquegraph;

   /* create tclique graph */
   SCIP_CALL( createTcliqueGraph(scip, &tcliquegraph) );

   /* define for each job a duration */
   SCIP_CALL( initializeDurations(scip, tcliquegraph, conss, nconss) );

   /* constuct incompatibility graph */
   SCIP_CALL( constructIncompatibilityGraph(scip, tcliquegraph, conss, nconss) );

   /* search for new precedence constraints */
   if( conshdlrdata->detectvarbounds )
   {
      SCIP_CALL( findPrecedenceConss(scip, tcliquegraph, naddconss) );
   }

   /* search for new cumulative constraints */
   if( conshdlrdata->detectdisjunctive )
   {
      SCIP_CALL( findCumulativeConss(scip, tcliquegraph, naddconss) );
   }

   /* free tclique graph data structure */
   freeTcliqueGraph(scip, &tcliquegraph);

   return SCIP_OKAY;
}

/** compute the constraint signature which is used to detect constraints which contain potentially the same set of variables */
static
void consdataCalcSignature(
   SCIP_CONSDATA*        consdata            /**< cumulative constraint data */
   )
{
   SCIP_VAR** vars;
   int nvars;
   int v;

   if( consdata->validsignature )
      return;

   vars = consdata->vars;
   nvars = consdata->nvars;

   for( v = 0; v < nvars; ++v )
   {
      consdata->signature |= ((unsigned int)1 << ((unsigned int)SCIPvarGetIndex(vars[v]) % (sizeof(unsigned int) * 8)));
   }

   consdata->validsignature = TRUE;
}

/** index comparison method of linear constraints: compares two indices of the variable set in the linear constraint */
static
SCIP_DECL_SORTINDCOMP(consdataCompVar)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata = (SCIP_CONSDATA*)dataptr;

   assert(consdata != NULL);
   assert(0 <= ind1 && ind1 < consdata->nvars);
   assert(0 <= ind2 && ind2 < consdata->nvars);

   return SCIPvarCompare(consdata->vars[ind1], consdata->vars[ind2]);
}

/** run a pairwise comparison */
static
SCIP_RETCODE removeRedundantConss(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           conss,              /**< array of cumulative constraints */
   int                   nconss,             /**< number of cumulative constraints */
   int*                  ndelconss           /**< pointer to store the number of deletedconstraints */
   )
{
   int i;
   int j;

   for( i = 0; i < nconss; ++i )
   {
      SCIP_CONSDATA* consdata0;
      SCIP_CONS* cons0;

      cons0 = conss[i];
      assert(cons0 != NULL);

      consdata0 = SCIPconsGetData(cons0);
      assert(consdata0 != NULL);

      consdataCalcSignature(consdata0);
      assert(consdata0->validsignature);

      for( j = i+1; j < nconss; ++j )
      {
         SCIP_CONSDATA* consdata1;
         SCIP_CONS* cons1;

         cons1 = conss[j];
         assert(cons1 != NULL);

         consdata1 = SCIPconsGetData(cons1);
         assert(consdata1 != NULL);

         if( consdata0->capacity != consdata1->capacity )
            continue;

         consdataCalcSignature(consdata1);
         assert(consdata1->validsignature);

         if( (consdata1->signature & (~consdata0->signature)) == 0 )
         {
            SCIPswapPointers((void**)&consdata0, (void**)&consdata1);
            SCIPswapPointers((void**)&cons0, (void**)&cons1);
            assert((consdata0->signature & (~consdata1->signature)) == 0);
         }

         if( (consdata0->signature & (~consdata1->signature)) == 0 )
         {
            int* perm0;
            int* perm1;
            int v0;
            int v1;

            if( consdata0->nvars > consdata1->nvars )
               continue;

            if( consdata0->hmin < consdata1->hmin )
               continue;

            if( consdata0->hmax > consdata1->hmax )
               continue;

            SCIP_CALL( SCIPallocBufferArray(scip, &perm0, consdata0->nvars) );
            SCIP_CALL( SCIPallocBufferArray(scip, &perm1, consdata1->nvars) );

            /* call sorting method  */
            SCIPsort(perm0, consdataCompVar, (void*)consdata0, consdata0->nvars);
            SCIPsort(perm1, consdataCompVar, (void*)consdata1, consdata1->nvars);

            for( v0 = 0, v1 = 0; v0 < consdata0->nvars && v1 < consdata1->nvars; )
            {
               SCIP_VAR* var0;
               SCIP_VAR* var1;
               int idx0;
               int idx1;
               int comp;

               idx0 = perm0[v0];
               idx1 = perm1[v1];

               var0 = consdata0->vars[idx0];

               var1 = consdata1->vars[idx1];

               comp = SCIPvarCompare(var0, var1);

               if( comp == 0 )
               {
                  int duration0;
                  int duration1;
                  int demand0;
                  int demand1;

                  demand0 = consdata0->demands[idx0];
                  duration0 = consdata0->durations[idx0];

                  demand1 = consdata1->demands[idx1];
                  duration1 = consdata1->durations[idx1];

                  if( demand0 != demand1 )
                     break;

                  if( duration0 != duration1 )
                     break;

                  v0++;
                  v1++;
               }
               else if( comp > 0 )
                  v1++;
               else
                  break;
            }

            if( v0 == consdata0->nvars )
            {
               if( SCIPconsIsChecked(cons0) && !SCIPconsIsChecked(cons1) )
               {
                  initializeLocks(consdata1, TRUE);
               }

               SCIP_CALL( SCIPupdateConsFlags(scip, cons1, cons0) );

               SCIP_CALL( SCIPdelCons(scip, cons0) );
               (*ndelconss)++;
            }

            SCIPfreeBufferArray(scip, &perm1);
            SCIPfreeBufferArray(scip, &perm0);
         }
      }
   }

   return SCIP_OKAY;
}

/** strengthen the variable bounds using the cumulative condition */
static
SCIP_RETCODE strengthenVarbounds(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint to propagate */
   int*                  nchgbds,            /**< pointer to store the number of changed bounds */
   int*                  naddconss           /**< pointer to store the number of added constraints */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_VAR** vars;
   int* durations;
   int* demands;
   int capacity;
   int nvars;
   int nconss;
   int i;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   /* check if the variable bounds got already strengthen by the cumulative constraint */
   if( consdata->varbounds )
      return SCIP_OKAY;

   vars = consdata->vars;
   durations = consdata->durations;
   demands = consdata->demands;
   capacity = consdata->capacity;
   nvars = consdata->nvars;

   nconss = 0;

   for( i = 0; i < nvars && !SCIPisStopped(scip); ++i )
   {
      SCIP_VAR** vbdvars;
      SCIP_VAR* var;
      SCIP_Real* vbdcoefs;
      SCIP_Real* vbdconsts;
      int nvbdvars;
      int b;
      int j;

      var = consdata->vars[i];
      assert(var != NULL);

      vbdvars = SCIPvarGetVlbVars(var);
      vbdcoefs = SCIPvarGetVlbCoefs(var);
      vbdconsts = SCIPvarGetVlbConstants(var);
      nvbdvars = SCIPvarGetNVlbs(var);

      for( b = 0; b < nvbdvars; ++b )
      {
         if( SCIPisEQ(scip, vbdcoefs[b], 1.0) )
         {
            if( SCIPconvertRealToInt(scip, vbdconsts[b]) > -durations[i] )
            {
               for( j = 0; j < nvars; ++j )
               {
                  if( vars[j] == vbdvars[b] )
                     break;
               }
               if( j == nvars )
                  continue;

               if( demands[i] +  demands[j] > capacity && SCIPconvertRealToInt(scip, vbdconsts[b]) < durations[j] )
               {
                  SCIP_Bool infeasible;
                  char name[SCIP_MAXSTRLEN];
                  int nlocalbdchgs;

                  SCIPdebugMessage("<%s>[%d] + %g <= <%s>[%d]\n", SCIPvarGetName(vbdvars[b]), durations[j], vbdconsts[b], SCIPvarGetName(var), durations[i]);

                  /* construct constraint name */
                  (void)SCIPsnprintf(name, SCIP_MAXSTRLEN, "varbound_%d_%d", SCIPgetNRuns(scip), nconss);

                  SCIP_CALL( createPrecedenceCons(scip, name, vars[j], vars[i], durations[j]) );
                  nconss++;

                  SCIP_CALL( SCIPaddVarVlb(scip, var, vbdvars[b], 1.0, (SCIP_Real) durations[j], &infeasible, &nlocalbdchgs) );
                  assert(!infeasible);

                  (*nchgbds) += nlocalbdchgs;
               }
            }
         }
      }
   }

   (*naddconss) += nconss;

   consdata->varbounds = TRUE;

   return SCIP_OKAY;
}

/**@} */


/**@name Callback methods of constraint handler
 *
 * @{
 */

/** copy method for constraint handler plugins (called when SCIP copies plugins) */
static
SCIP_DECL_CONSHDLRCOPY(conshdlrCopyCumulative)
{  /*lint --e{715}*/
   assert(scip != NULL);
   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);

   /* call inclusion method of constraint handler */
   SCIP_CALL( SCIPincludeConshdlrCumulative(scip) );

   SCIPstatistic( SCIPconshdlrGetData(SCIPfindConshdlr(scip, CONSHDLR_NAME))->iscopy = TRUE );

   *valid = TRUE;

   return SCIP_OKAY;
}

/** destructor of constraint handler to free constraint handler data (called when SCIP is exiting) */
static
SCIP_DECL_CONSFREE(consFreeCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

#ifdef SCIP_STATISTIC
   if( !conshdlrdata->iscopy )
   {
      /* statisitc output if SCIP_STATISTIC is defined */
      SCIPstatisticPrintf("time-table: lb=%" SCIP_LONGINT_FORMAT ", ub=%" SCIP_LONGINT_FORMAT ", cutoff=%" SCIP_LONGINT_FORMAT "\n",
         conshdlrdata->nlbtimetable, conshdlrdata->nubtimetable, conshdlrdata->ncutofftimetable);
      SCIPstatisticPrintf("edge-finder: lb=%" SCIP_LONGINT_FORMAT ", ub=%" SCIP_LONGINT_FORMAT ", cutoff=%" SCIP_LONGINT_FORMAT "\n",
         conshdlrdata->nlbedgefinder, conshdlrdata->nubedgefinder, conshdlrdata->ncutoffedgefinder);
      SCIPstatisticPrintf("overload: time-table=%" SCIP_LONGINT_FORMAT " time-time edge-finding=%" SCIP_LONGINT_FORMAT "\n",
      conshdlrdata->ncutoffoverload, conshdlrdata->ncutoffoverloadTTEF);
   }
#endif

   conshdlrdataFree(scip, &conshdlrdata);

   SCIPconshdlrSetData(conshdlr, NULL);

   return SCIP_OKAY;
}


/** presolving initialization method of constraint handler (called when presolving is about to begin) */
static
SCIP_DECL_CONSINITPRE(consInitpreCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   int c;

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   conshdlrdata->detectedredundant = FALSE;

   for( c = 0; c < nconss; ++c )
   {
      /* remove jobs which have a duration or demand of zero (zero energy) or lay outside the effective horizon [hmin,
       * hmax)
       */
      SCIP_CALL( removeIrrelevantJobs(scip, conss[c]) );
   }

   return SCIP_OKAY;
}


/** presolving deinitialization method of constraint handler (called after presolving has been finished) */
#ifdef SCIP_STATISTIC
static
SCIP_DECL_CONSEXITPRE(consExitpreCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   int c;

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   for( c = 0; c < nconss; ++c )
   {
      SCIP_CALL( evaluateCumulativeness(scip, conss[c]) );

#if 0
      SCIP_CALL( SCIPvisualizeConsCumulative(scip, conss[c]) );
#endif
   }

   if( !conshdlrdata->iscopy )
   {
      SCIPstatisticPrintf("@11  added variables bounds constraints %d\n", conshdlrdata->naddedvarbounds);
      SCIPstatisticPrintf("@22  added disjunctive constraints %d\n", conshdlrdata->naddeddisjunctives);
      SCIPstatisticPrintf("@33  irrelevant %d\n", conshdlrdata->nirrelevantjobs);
      SCIPstatisticPrintf("@44  dual %d\n", conshdlrdata->ndualfixs);
      SCIPstatisticPrintf("@55  locks %d\n", conshdlrdata->nremovedlocks);
      SCIPstatisticPrintf("@66  decomp %d\n", conshdlrdata->ndecomps);
      SCIPstatisticPrintf("@77  allconsdual %d\n", conshdlrdata->nallconsdualfixs);
      SCIPstatisticPrintf("@88  alwaysruns %d\n", conshdlrdata->nalwaysruns);
      SCIPstatisticPrintf("@99  dualbranch %d\n", conshdlrdata->ndualbranchs);
   }

   return SCIP_OKAY;
}
#endif


/** solving process deinitialization method of constraint handler (called before branch and bound process data is freed) */
static
SCIP_DECL_CONSEXITSOL(consExitsolCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;
   int c;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);

   /* release the rows of all constraints */
   for( c = 0; c < nconss; ++c )
   {
      consdata = SCIPconsGetData(conss[c]);
      assert(consdata != NULL);

      /* free rows */
      SCIP_CALL( consdataFreeRows(scip, &consdata) );
   }

   return SCIP_OKAY;
}

/** frees specific constraint data */
static
SCIP_DECL_CONSDELETE(consDeleteCumulative)
{  /*lint --e{715}*/
   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(consdata != NULL );
   assert(*consdata != NULL );

   /* if constraint belongs to transformed problem space, drop bound change events on variables */
   if( (*consdata)->nvars > 0 && SCIPvarIsTransformed((*consdata)->vars[0]) )
   {
      SCIP_CONSHDLRDATA* conshdlrdata;

      conshdlrdata = SCIPconshdlrGetData(conshdlr);
      assert(conshdlrdata != NULL);

      SCIP_CALL( consdataDropAllEvents(scip, *consdata, conshdlrdata->eventhdlr) );
   }

   /* free cumulative constraint data */
   SCIP_CALL( consdataFree(scip, consdata) );

   return SCIP_OKAY;
}

/** transforms constraint data into data belonging to the transformed problem */
static
SCIP_DECL_CONSTRANS(consTransCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_CONSDATA* sourcedata;
   SCIP_CONSDATA* targetdata;

   assert(conshdlr != NULL);
   assert(SCIPgetStage(scip) == SCIP_STAGE_TRANSFORMING);
   assert(sourcecons != NULL);
   assert(targetcons != NULL);

   sourcedata = SCIPconsGetData(sourcecons);
   assert(sourcedata != NULL);
   assert(sourcedata->demandrows == NULL);

   SCIPdebugMessage("transform cumulative constraint <%s>\n", SCIPconsGetName(sourcecons));

   /* get event handler */
   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);
   assert(conshdlrdata->eventhdlr != NULL);

   /* create constraint data for target constraint */
   SCIP_CALL( consdataCreate(scip, &targetdata, sourcedata->vars, sourcedata->linkingconss,
         sourcedata->durations, sourcedata->demands, sourcedata->nvars, sourcedata->capacity,
         sourcedata->hmin, sourcedata->hmax, SCIPconsIsChecked(sourcecons)) );

   /* create target constraint */
   SCIP_CALL( SCIPcreateCons(scip, targetcons, SCIPconsGetName(sourcecons), conshdlr, targetdata,
         SCIPconsIsInitial(sourcecons), SCIPconsIsSeparated(sourcecons), SCIPconsIsEnforced(sourcecons),
         SCIPconsIsChecked(sourcecons), SCIPconsIsPropagated(sourcecons),
         SCIPconsIsLocal(sourcecons), SCIPconsIsModifiable(sourcecons),
         SCIPconsIsDynamic(sourcecons), SCIPconsIsRemovable(sourcecons), SCIPconsIsStickingAtNode(sourcecons)) );

   /* catch bound change events of variables */
   SCIP_CALL( consdataCatchEvents(scip, targetdata, conshdlrdata->eventhdlr) );

   return SCIP_OKAY;
}

/** LP initialization method of constraint handler */
static
SCIP_DECL_CONSINITLP(consInitlpCumulative)
{
   SCIP_CONSHDLRDATA* conshdlrdata;
   int c;

   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(conshdlr != NULL);
   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   SCIPdebugMessage("initialize LP relaxation for %d cumulative constraints\n", nconss);

   if( conshdlrdata->usebinvars )
   {
      /* add rows to LP */
      for( c = 0; c < nconss; ++c )
      {
         assert(SCIPconsIsInitial(conss[c]));
         SCIP_CALL( addRelaxation(scip, conss[c], conshdlrdata->cutsasconss) );

         if( conshdlrdata->cutsasconss )
         {
            SCIP_CALL( SCIPrestartSolve(scip) );
         }
      }
   }

   /**@todo if we want to use only the integer variables; only these will be in cuts
    *       create some initial cuts, currently these are only separated */

   return SCIP_OKAY;
}

/** separation method of constraint handler for LP solutions */
static
SCIP_DECL_CONSSEPALP(consSepalpCumulative)
{
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_Bool cutoff;
   SCIP_Bool reducedom;
   SCIP_Bool separated;
   int c;

   SCIPdebugMessage("consSepalpCumulative\n");

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);
   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   SCIPdebugMessage("separating %d/%d cumulative constraints\n", nusefulconss, nconss);

   cutoff = FALSE;
   reducedom = FALSE;
   separated = FALSE;
   (*result) = SCIP_DIDNOTRUN;

   if( !conshdlrdata->localcuts && SCIPgetDepth(scip) > 0 )
      return SCIP_OKAY;

   (*result) = SCIP_DIDNOTFIND;

   if( conshdlrdata->usebinvars )
   {
      /* check all useful cumulative constraints for feasibility  */
      for( c = 0; c < nusefulconss && !reducedom && !cutoff; ++c )
      {
         SCIP_CALL( separateConsBinaryRepresentation(scip, conss[c], NULL, &separated, &cutoff) );
      }

      if( !cutoff && !reducedom && conshdlrdata->usecovercuts )
      {
         for( c = 0; c < nusefulconss; ++c )
         {
            SCIP_CALL( separateCoverCutsCons(scip, conss[c], NULL, &separated, &cutoff) );
         }
      }
   }

   if( conshdlrdata->sepaold )
   {
      /* separate cuts containing only integer variables */
      for( c = 0; c < nusefulconss; ++c )
      {
         SCIP_CALL( separateConsOnIntegerVariables(scip, conss[c], NULL, TRUE, &separated) );
         SCIP_CALL( separateConsOnIntegerVariables(scip, conss[c], NULL, FALSE, &separated) );
      }
   }

   if( cutoff )
      *result = SCIP_CUTOFF;
   else if( reducedom )
      *result = SCIP_REDUCEDDOM;
   else if( separated )
      *result = SCIP_SEPARATED;

   return SCIP_OKAY;
}

/** separation method of constraint handler for arbitrary primal solutions */
static
SCIP_DECL_CONSSEPASOL(consSepasolCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_Bool cutoff;
   SCIP_Bool reducedom;
   SCIP_Bool separated;
   int c;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   if( !conshdlrdata->localcuts && SCIPgetDepth(scip) > 0 )
      return SCIP_OKAY;

   SCIPdebugMessage("separating %d/%d cumulative constraints\n", nusefulconss, nconss);

   cutoff = FALSE;
   reducedom = FALSE;
   separated = FALSE;
   (*result) = SCIP_DIDNOTFIND;

   if( conshdlrdata->usebinvars )
   {
      /* check all useful cumulative constraints for feasibility  */
      for( c = 0; c < nusefulconss && !cutoff && !reducedom; ++c )
      {
         SCIP_CALL( separateConsBinaryRepresentation(scip, conss[c], NULL, &separated, &cutoff) );
      }

      if( !cutoff && !reducedom && conshdlrdata->usecovercuts )
      {
         for( c = 0; c < nusefulconss; ++c )
         {
            SCIP_CALL( separateCoverCutsCons(scip, conss[c], sol, &separated, &cutoff) );
         }
      }
   }
   if( conshdlrdata->sepaold )
   {
      /* separate cuts containing only integer variables */
      for( c = 0; c < nusefulconss; ++c )
      {
         SCIP_CALL( separateConsOnIntegerVariables(scip, conss[c], NULL, TRUE, &separated) );
         SCIP_CALL( separateConsOnIntegerVariables(scip, conss[c], NULL, FALSE, &separated) );
      }
   }

   if( cutoff )
      *result = SCIP_CUTOFF;
   else if( reducedom )
      *result = SCIP_REDUCEDDOM;
   else if( separated )
      *result = SCIP_SEPARATED;

   return SCIP_OKAY;
}

/** constraint enforcing method of constraint handler for LP solutions */
static
SCIP_DECL_CONSENFOLP(consEnfolpCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);

   if( solinfeasible )
   {
      *result = SCIP_INFEASIBLE;
      return SCIP_OKAY;
   }

   SCIPdebugMessage("LP enforcing %d useful cumulative constraints of %d constraints\n", nusefulconss, nconss);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   (*result) = SCIP_FEASIBLE;

   if( conshdlrdata->usebinvars )
   {
      SCIP_Bool separated;
      SCIP_Bool cutoff;
      int c;

      separated = FALSE;

      /* first check if a constraints is violated */
      for( c = 0; c < nusefulconss; ++c )
      {
         SCIP_CONS* cons;
         SCIP_Bool violated;

         cons = conss[c];
         assert(cons != NULL);

         SCIP_CALL( checkCons(scip, cons, NULL, &violated, FALSE) );

         if( !violated )
            continue;

         SCIP_CALL( separateConsBinaryRepresentation(scip, cons, NULL, &separated, &cutoff) );
         if ( cutoff )
         {
            *result = SCIP_CUTOFF;
            return SCIP_OKAY;
         }
      }

      for( ; c < nconss && !separated; ++c )
      {
         SCIP_CONS* cons;
         SCIP_Bool violated;

         cons = conss[c];
         assert(cons != NULL);

         SCIP_CALL( checkCons(scip, cons, NULL, &violated, FALSE) );

         if( !violated )
            continue;

         SCIP_CALL( separateConsBinaryRepresentation(scip, cons, NULL, &separated, &cutoff) );
         if ( cutoff )
         {
            *result = SCIP_CUTOFF;
            return SCIP_OKAY;
         }
      }

      if( separated )
         (*result) = SCIP_SEPARATED;
   }
   else
   {
      SCIP_CALL( enforceSolution(scip, conss, nconss, conshdlrdata->fillbranchcands, result) );
   }

   return SCIP_OKAY;
}

/** constraint enforcing method of constraint handler for pseudo solutions */
static
SCIP_DECL_CONSENFOPS(consEnfopsCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;

   SCIPdebugMessage("method: enforce pseudo solution\n");

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);

   if( objinfeasible )
   {
      *result = SCIP_DIDNOTRUN;
      return SCIP_OKAY;
   }

   (*result) = SCIP_FEASIBLE;

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   SCIP_CALL( enforceSolution(scip, conss, nconss, conshdlrdata->fillbranchcands, result) );

   return SCIP_OKAY;
}

/** feasibility check method of constraint handler for integral solutions */
static
SCIP_DECL_CONSCHECK(consCheckCumulative)
{  /*lint --e{715}*/
   SCIP_Bool violated;
   int c;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);

   violated = FALSE;

   SCIPdebugMessage("check %d cumulative constraints\n", nconss);

   for( c = 0; c < nconss && !violated; ++c )
   {
      SCIP_CALL( checkCons(scip, conss[c], sol, &violated, printreason) );
   }

   if( violated )
      *result = SCIP_INFEASIBLE;
   else
      *result = SCIP_FEASIBLE;

   return SCIP_OKAY;
}

/** domain propagation method of constraint handler */
static
SCIP_DECL_CONSPROP(consPropCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_Bool cutoff;
   int nchgbds;
   int ndelconss;
   int c;
#if 0
   int naggrvars = 0;
#endif

   SCIPdebugMessage("propagate %d of %d useful cumulative constraints\n", nusefulconss, nconss);

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(nconss == 0 || conss != NULL);
   assert(result != NULL);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   nchgbds = 0;
   ndelconss = 0;
   cutoff = FALSE;
   (*result) = SCIP_DIDNOTRUN;

   /* propgate all useful constraints */
   for( c = 0; c < nusefulconss && !cutoff; ++c )
   {
      SCIP_CONS* cons;

      cons = conss[c];
      assert(cons != NULL);

      if( SCIPgetDepth(scip) == 0 )
      {
#if 0
         SCIP_CALL( presolveCons(scip, cons, conshdlrdata, SCIP_PRESOLTIMING_ALWAYS,
               &nchgbds, &naggrvars, &nchgbds, &ndelconss, &nchgbds, &nchgbds, &nchgbds, &cutoff, &cutoff) );
#else
         SCIP_CALL( presolveCons(scip, cons, conshdlrdata, SCIP_PRESOLTIMING_ALWAYS,
               &nchgbds, &nchgbds, &ndelconss, &nchgbds, &nchgbds, &nchgbds, &cutoff, &cutoff) );
#endif
         if( cutoff )
            break;

         if( SCIPconsIsDeleted(cons) )
            continue;
      }

      SCIP_CALL( propagateCons(scip, cons, conshdlrdata, SCIP_PRESOLTIMING_ALWAYS, &nchgbds, &ndelconss, &cutoff) );
   }

   if( !cutoff && nchgbds == 0 )
   {
      /* propgate all other constraints */
      for( c = nusefulconss; c < nconss && !cutoff; ++c )
      {
         SCIP_CALL( propagateCons(scip, conss[c], conshdlrdata, SCIP_PRESOLTIMING_ALWAYS, &nchgbds, &ndelconss, &cutoff) );
      }
   }

#if 0
   if( !cutoff && conshdlrdata->dualpresolve && SCIPallowDualReds(scip) && nconss > 1 )
   {
      SCIP_CALL( propagateAllConss(scip, conshdlrdata, conss, nconss, TRUE, &nchgbds, &cutoff, NULL) );
   }
#endif

   if( cutoff )
   {
      SCIPdebugMessage("detected infeasible\n");
      *result = SCIP_CUTOFF;
   }
   else if( nchgbds > 0 )
   {
      SCIPdebugMessage("delete (locally) %d constraints and changed %d variable bounds\n", ndelconss, nchgbds);
      *result = SCIP_REDUCEDDOM;
   }
   else
      *result = SCIP_DIDNOTFIND;

   return SCIP_OKAY;
}

/** presolving method of constraint handler */
static
SCIP_DECL_CONSPRESOL(consPresolCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_CONS* cons;
   SCIP_Bool cutoff;
   SCIP_Bool unbounded;
   int oldnfixedvars;
   int oldnchgbds;
   int oldndelconss;
   int oldnaddconss;
   int oldnupgdconss;
   int oldnchgsides;
   int oldnchgcoefs;
   int c;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(scip != NULL);
   assert(result != NULL);

   SCIPdebugMessage("presolve %d cumulative constraints\n", nconss);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   *result = SCIP_DIDNOTRUN;

   oldnfixedvars = *nfixedvars;
   oldnchgbds = *nchgbds;
   oldnchgsides = *nchgsides;
   oldnchgcoefs = *nchgcoefs;
   oldnupgdconss = *nupgdconss;
   oldndelconss = *ndelconss;
   oldnaddconss = *naddconss;
   cutoff = FALSE;
   unbounded = FALSE;

   /* process constraints */
   for( c = 0; c < nconss && !cutoff; ++c )
   {
      cons = conss[c];

      /* remove jobs which have a duration or demand of zero (zero energy) or lay outside the effective horizon [hmin,
       * hmax)
       */
      SCIP_CALL( removeIrrelevantJobs(scip, conss[c]) );

      if( presoltiming != SCIP_PRESOLTIMING_MEDIUM )
      {
#if 0
         SCIP_CALL( presolveCons(scip, cons, conshdlrdata, presoltiming,
               nfixedvars, naggrvars, nchgbds, ndelconss, naddconss, nchgcoefs, nchgsides, &cutoff, &unbounded) );
#else
         SCIP_CALL( presolveCons(scip, cons, conshdlrdata, presoltiming,
               nfixedvars, nchgbds, ndelconss, naddconss, nchgcoefs, nchgsides, &cutoff, &unbounded) );
#endif

         if( cutoff || unbounded )
            break;

         if( SCIPconsIsDeleted(cons) )
            continue;
      }

      /* in the first round we create a disjunctive constraint containing those jobs which cannot run in parallel */
      if( nrounds == 1 && SCIPgetNRuns(scip) == 1 && conshdlrdata->disjunctive )
      {
         SCIP_CALL( createDisjuctiveCons(scip, cons, naddconss) );
      }

      /* strengthen existing variable bounds using the cumulative condition */
      if( (presoltiming & SCIP_PRESOLTIMING_MEDIUM) != 0 )
      {
         SCIP_CALL( strengthenVarbounds(scip, cons, nchgbds, naddconss) );
      }

      /* propagate cumulative constraint */
      SCIP_CALL( propagateCons(scip, cons, conshdlrdata, presoltiming, nchgbds, ndelconss, &cutoff) );
      assert(checkDemands(scip, cons) || cutoff);
   }

   if( !cutoff && !unbounded && conshdlrdata->dualpresolve && SCIPallowDualReds(scip) && nconss > 1 && (presoltiming & SCIP_PRESOLTIMING_FAST) != 0 )
   {
      SCIP_CALL( propagateAllConss(scip, conshdlrdata, conss, nconss, FALSE,
            nfixedvars, &cutoff, NULL) );
   }

   /* only perform the detection of variable bounds and disjunctive constraint once */
   if( !cutoff && SCIPgetNRuns(scip) == 1 && !conshdlrdata->detectedredundant
      && (conshdlrdata->detectvarbounds || conshdlrdata->detectdisjunctive)
      && (presoltiming & SCIP_PRESOLTIMING_EXHAUSTIVE) != 0 )
   {
      /* combine different source and detect disjunctive constraints and variable bound constraints to improve the
       * propagation
       */
      SCIP_CALL( detectRedundantConss(scip, conshdlrdata, conss, nconss, naddconss) );
      conshdlrdata->detectedredundant = TRUE;
   }

   if( !cutoff && conshdlrdata->presolpairwise && (presoltiming & SCIP_PRESOLTIMING_MEDIUM) != 0 )
   {
      SCIP_CALL( removeRedundantConss(scip, conss, nconss, ndelconss) );
   }

   SCIPdebugMessage("delete %d constraints and changed %d variable bounds (cutoff %u)\n",
      *ndelconss - oldndelconss, *nchgbds - oldnchgbds, cutoff);

   if( cutoff )
      *result = SCIP_CUTOFF;
   else if( unbounded )
      *result = SCIP_UNBOUNDED;
   else if( *nchgbds > oldnchgbds || *nfixedvars > oldnfixedvars || *nchgsides > oldnchgsides
      || *nchgcoefs > oldnchgcoefs  || *nupgdconss > oldnupgdconss || *ndelconss > oldndelconss || *naddconss > oldnaddconss )
      *result = SCIP_SUCCESS;
   else
      *result = SCIP_DIDNOTFIND;

   return SCIP_OKAY;
}

/** propagation conflict resolving method of constraint handler */
static
SCIP_DECL_CONSRESPROP(consRespropCumulative)
{  /*lint --e{715}*/
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_CONSDATA* consdata;

   assert(conshdlr != NULL);
   assert(strcmp(SCIPconshdlrGetName(conshdlr), CONSHDLR_NAME) == 0);
   assert(scip != NULL);
   assert(result != NULL);
   assert(infervar != NULL);
   assert(bdchgidx != NULL);

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   /* process constraint */
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   SCIPdebugMessage("resolve propagation: variable <%s>, cumulative constraint <%s> (capacity %d, propagation %d, H=[%d,%d))\n",
      SCIPvarGetName(infervar), SCIPconsGetName(cons), consdata->capacity, inferInfoGetProprule(intToInferInfo(inferinfo)),
      SCIPgetHminCumulative(scip, cons), SCIPgetHmaxCumulative(scip, cons));

   SCIP_CALL( respropCumulativeCondition(scip, consdata->nvars, consdata->vars,
         consdata->durations, consdata->demands, consdata->capacity, consdata->hmin, consdata->hmax,
         infervar, intToInferInfo(inferinfo), boundtype, bdchgidx, relaxedbd, conshdlrdata->usebdwidening, NULL, result) );

   return SCIP_OKAY;
}

/** variable rounding lock method of constraint handler */
static
SCIP_DECL_CONSLOCK(consLockCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;
   SCIP_VAR** vars;
   int v;

   SCIPdebugMessage("lock cumulative constraint <%s> with nlockspos = %d, nlocksneg = %d\n", SCIPconsGetName(cons), nlockspos, nlocksneg);

   assert(scip != NULL);
   assert(cons != NULL);

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   vars = consdata->vars;
   assert(vars != NULL);

   for( v = 0; v < consdata->nvars; ++v )
   {
      if( consdata->downlocks[v] && consdata->uplocks[v] )
      {
         /* the integer start variable should not get rounded in both direction  */
         SCIP_CALL( SCIPaddVarLocks(scip, vars[v], nlockspos + nlocksneg, nlockspos + nlocksneg) );
      }
      else if( consdata->downlocks[v]  )
      {
         SCIP_CALL( SCIPaddVarLocks(scip, vars[v], nlockspos, nlocksneg) );
      }
      else if( consdata->uplocks[v] )
      {
         SCIP_CALL( SCIPaddVarLocks(scip, vars[v], nlocksneg, nlockspos) );
      }
   }

   return SCIP_OKAY;
}


/** constraint display method of constraint handler */
static
SCIP_DECL_CONSPRINT(consPrintCumulative)
{  /*lint --e{715}*/
   assert(scip != NULL);
   assert(conshdlr != NULL);
   assert(cons != NULL);

   consdataPrint(scip, SCIPconsGetData(cons), file);

   return SCIP_OKAY;
}

/** constraint copying method of constraint handler */
static
SCIP_DECL_CONSCOPY(consCopyCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* sourceconsdata;
   SCIP_VAR** sourcevars;
   SCIP_VAR** vars;
   const char* consname;

   int nvars;
   int v;

   sourceconsdata = SCIPconsGetData(sourcecons);
   assert(sourceconsdata != NULL);

   /* get variables of the source constraint */
   nvars = sourceconsdata->nvars;
   sourcevars = sourceconsdata->vars;

   (*valid) = TRUE;

   if( nvars == 0 )
      return SCIP_OKAY;

   /* allocate buffer array */
   SCIP_CALL( SCIPallocBufferArray(scip, &vars, nvars) );

   for( v = 0; v < nvars && *valid; ++v )
   {
      SCIP_CALL( SCIPgetVarCopy(sourcescip, scip, sourcevars[v], &vars[v], varmap, consmap, global, valid) );
      assert(!(*valid) || vars[v] != NULL);
   }

   /* only create the target constraint, if all variables could be copied */
   if( *valid )
   {
      if( name != NULL )
         consname = name;
      else
         consname = SCIPconsGetName(sourcecons);

      /* create a copy of the cumulative constraint */
      SCIP_CALL( SCIPcreateConsCumulative(scip, cons, consname, nvars, vars,
            sourceconsdata->durations, sourceconsdata->demands, sourceconsdata->capacity,
            initial, separate, enforce, check, propagate, local, modifiable, dynamic, removable, stickingatnode) );

      /* adjust left side if the time axis if needed */
      if( sourceconsdata->hmin > 0 )
      {
         SCIP_CALL( SCIPsetHminCumulative(scip, *cons, sourceconsdata->hmin) );
      }

      /* adjust right  side if the time axis if needed */
      if( sourceconsdata->hmax < INT_MAX )
      {
         SCIP_CALL( SCIPsetHmaxCumulative(scip, *cons, sourceconsdata->hmax) );
      }
   }

   /* free buffer array */
   SCIPfreeBufferArray(scip, &vars);

   return SCIP_OKAY;
}


/** constraint parsing method of constraint handler */
static
SCIP_DECL_CONSPARSE(consParseCumulative)
{  /*lint --e{715}*/
   SCIP_VAR** vars;
   SCIP_VAR* var;
   SCIP_Real value;
   char strvalue[SCIP_MAXSTRLEN];
   char* endptr;
   int* demands;
   int* durations;
   int capacity;
   int duration;
   int demand;
   int hmin;
   int hmax;
   int varssize;
   int nvars;

   SCIPdebugMessage("parse <%s> as cumulative constraint\n", str);

   /* cutoff "cumulative" form the constraint string */
   SCIPstrCopySection(str, 'c', '(', strvalue, SCIP_MAXSTRLEN, &endptr);
   str = endptr;

   varssize = 100;
   nvars = 0;

   /* allocate buffer array for variables */
   SCIP_CALL( SCIPallocBufferArray(scip, &vars, varssize) );
   SCIP_CALL( SCIPallocBufferArray(scip, &demands, varssize) );
   SCIP_CALL( SCIPallocBufferArray(scip, &durations, varssize) );

   do
   {
      SCIP_CALL( SCIPparseVarName(scip, str, &var, &endptr) );

      if( var != NULL )
      {
         str = endptr;

         SCIPstrCopySection(str, '(', ')', strvalue, SCIP_MAXSTRLEN, &endptr);
         duration = atoi(strvalue);
         str = endptr;

         SCIPstrCopySection(str, '[', ']', strvalue, SCIP_MAXSTRLEN, &endptr);
         demand = atoi(strvalue);
         str = endptr;

         SCIPdebugMessage("parse job <%s>, duration %d, demand %d\n", SCIPvarGetName(var), duration, demand);

         vars[nvars] = var;
         demands[nvars] = demand;
         durations[nvars] = duration;
         nvars++;
      }
   }
   while( var != NULL );

   /* parse effective time window */
   SCIPstrCopySection(str, '[', ',', strvalue, SCIP_MAXSTRLEN, &endptr);
   hmin = atoi(strvalue);
   str = endptr;

   if( SCIPstrToRealValue(str, &value, &endptr) )
   {
      hmax = (int)(value);
      str = endptr;

      /* parse capacity */
      SCIPstrCopySection(str, ')', '=', strvalue, SCIP_MAXSTRLEN, &endptr);
      str = endptr;
      if( SCIPstrToRealValue(str, &value, &endptr) )
      {
         capacity = (int)value;

         /* create cumulative constraint */
         SCIP_CALL( SCIPcreateConsCumulative(scip, cons, name, nvars, vars, durations, demands, capacity,
               initial, separate, enforce, check, propagate, local, modifiable, dynamic, removable, stickingatnode) );

         SCIP_CALL( SCIPsetHminCumulative(scip, *cons, hmin) );
         SCIP_CALL( SCIPsetHmaxCumulative(scip, *cons, hmax) );

         (*success) = TRUE;
      }
   }

   /* free buffer arrays */
   SCIPfreeBufferArray(scip, &durations);
   SCIPfreeBufferArray(scip, &demands);
   SCIPfreeBufferArray(scip, &vars);

   return SCIP_OKAY;
}

/** constraint method of constraint handler which returns the variables (if possible) */
static
SCIP_DECL_CONSGETVARS(consGetVarsCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   if( varssize < consdata->nvars )
      (*success) = FALSE;
   else
   {
      assert(vars != NULL);

      BMScopyMemoryArray(vars, consdata->vars, consdata->nvars);
      (*success) = TRUE;
   }

   return SCIP_OKAY;
}

/** constraint method of constraint handler which returns the number of variables (if possible) */
static
SCIP_DECL_CONSGETNVARS(consGetNVarsCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   (*nvars) = consdata->nvars;
   (*success) = TRUE;

   return SCIP_OKAY;
}

/**@} */

/**@name Callback methods of event handler
 *
 * @{
 */


/** execution method of event handler */
static
SCIP_DECL_EVENTEXEC(eventExecCumulative)
{  /*lint --e{715}*/
   SCIP_CONSDATA* consdata;

   assert(scip != NULL);
   assert(eventhdlr != NULL);
   assert(eventdata != NULL);
   assert(strcmp(SCIPeventhdlrGetName(eventhdlr), EVENTHDLR_NAME) == 0);
   assert(event != NULL);

   consdata = (SCIP_CONSDATA*)eventdata;
   assert(consdata != NULL);

   /* mark the constraint to be not propagated */
   consdata->propagated = FALSE;

   return SCIP_OKAY;
}

/**@} */

/**@name Interface methods
 *
 * @{
 */

/*
 * constraint specific interface methods
 */

/** creates the handler for cumulative constraints and includes it in SCIP */
SCIP_RETCODE SCIPincludeConshdlrCumulative(
   SCIP*                 scip                /**< SCIP data structure */
   )
{
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_CONSHDLR* conshdlr;
   SCIP_EVENTHDLR* eventhdlr;

   /* create event handler for bound change events */
   SCIP_CALL( SCIPincludeEventhdlrBasic(scip, &eventhdlr, EVENTHDLR_NAME, EVENTHDLR_DESC, eventExecCumulative, NULL) );

   /* create cumulative constraint handler data */
   SCIP_CALL( conshdlrdataCreate(scip, &conshdlrdata, eventhdlr) );

   /* include constraint handler */
   SCIP_CALL( SCIPincludeConshdlrBasic(scip, &conshdlr, CONSHDLR_NAME, CONSHDLR_DESC,
         CONSHDLR_ENFOPRIORITY, CONSHDLR_CHECKPRIORITY, CONSHDLR_EAGERFREQ, CONSHDLR_NEEDSCONS,
         consEnfolpCumulative, consEnfopsCumulative, consCheckCumulative, consLockCumulative,
         conshdlrdata) );

   assert(conshdlr != NULL);

   /* set non-fundamental callbacks via specific setter functions */
   SCIP_CALL( SCIPsetConshdlrCopy(scip, conshdlr, conshdlrCopyCumulative, consCopyCumulative) );
   SCIP_CALL( SCIPsetConshdlrDelete(scip, conshdlr, consDeleteCumulative) );
#ifdef SCIP_STATISTIC
   SCIP_CALL( SCIPsetConshdlrExitpre(scip, conshdlr, consExitpreCumulative) );
#endif
   SCIP_CALL( SCIPsetConshdlrExitsol(scip, conshdlr, consExitsolCumulative) );
   SCIP_CALL( SCIPsetConshdlrFree(scip, conshdlr, consFreeCumulative) );
   SCIP_CALL( SCIPsetConshdlrGetVars(scip, conshdlr, consGetVarsCumulative) );
   SCIP_CALL( SCIPsetConshdlrGetNVars(scip, conshdlr, consGetNVarsCumulative) );
   SCIP_CALL( SCIPsetConshdlrInitpre(scip, conshdlr, consInitpreCumulative) );
   SCIP_CALL( SCIPsetConshdlrInitlp(scip, conshdlr, consInitlpCumulative) );
   SCIP_CALL( SCIPsetConshdlrParse(scip, conshdlr, consParseCumulative) );
   SCIP_CALL( SCIPsetConshdlrPresol(scip, conshdlr, consPresolCumulative, CONSHDLR_MAXPREROUNDS,
         CONSHDLR_PRESOLTIMING) );
   SCIP_CALL( SCIPsetConshdlrPrint(scip, conshdlr, consPrintCumulative) );
   SCIP_CALL( SCIPsetConshdlrProp(scip, conshdlr, consPropCumulative, CONSHDLR_PROPFREQ, CONSHDLR_DELAYPROP,
         CONSHDLR_PROP_TIMING) );
   SCIP_CALL( SCIPsetConshdlrResprop(scip, conshdlr, consRespropCumulative) );
   SCIP_CALL( SCIPsetConshdlrSepa(scip, conshdlr, consSepalpCumulative, consSepasolCumulative, CONSHDLR_SEPAFREQ,
         CONSHDLR_SEPAPRIORITY, CONSHDLR_DELAYSEPA) );
   SCIP_CALL( SCIPsetConshdlrTrans(scip, conshdlr, consTransCumulative) );

   /* add cumulative constraint handler parameters */
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/ttinfer",
         "should time-table (core-times) propagator be used to infer bounds?",
         &conshdlrdata->ttinfer, FALSE, DEFAULT_TTINFER, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/efcheck",
         "should edge-finding be used to detect an overload?",
         &conshdlrdata->efcheck, FALSE, DEFAULT_EFCHECK, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/efinfer",
         "should edge-finding be used to infer bounds?",
         &conshdlrdata->efinfer, FALSE, DEFAULT_EFINFER, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/useadjustedjobs", "should edge-finding be executed?",
         &conshdlrdata->useadjustedjobs, TRUE, DEFAULT_USEADJUSTEDJOBS, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/ttefcheck",
         "should time-table edge-finding be used to detect an overload?",
         &conshdlrdata->ttefcheck, FALSE, DEFAULT_TTEFCHECK, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/ttefinfer",
         "should time-table edge-finding be used to infer bounds?",
         &conshdlrdata->ttefinfer, FALSE, DEFAULT_TTEFINFER, NULL, NULL) );

   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/usebinvars", "should the binary representation be used?",
         &conshdlrdata->usebinvars, FALSE, DEFAULT_USEBINVARS, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/localcuts", "should cuts be added only locally?",
         &conshdlrdata->localcuts, FALSE, DEFAULT_LOCALCUTS, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/usecovercuts", "should covering cuts be added every node?",
         &conshdlrdata->usecovercuts, FALSE, DEFAULT_USECOVERCUTS, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/cutsasconss",
         "should the cumulative constraint create cuts as knapsack constraints?",
         &conshdlrdata->cutsasconss, FALSE, DEFAULT_CUTSASCONSS, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/sepaold",
         "shall old sepa algo be applied?",
         &conshdlrdata->sepaold, FALSE, DEFAULT_SEPAOLD, NULL, NULL) );

   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/fillbranchcands", "should branching candidates be added to storage?",
         &conshdlrdata->fillbranchcands, FALSE, DEFAULT_FILLBRANCHCANDS, NULL, NULL) );

   /* presolving parameters */
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/dualpresolve", "should dual presolving be applied?",
         &conshdlrdata->dualpresolve, FALSE, DEFAULT_DUALPRESOLVE, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/coeftightening", "should coefficient tightening be applied?",
         &conshdlrdata->coeftightening, FALSE, DEFAULT_COEFTIGHTENING, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/normalize", "should demands and capacity be normalized?",
         &conshdlrdata->normalize, FALSE, DEFAULT_NORMALIZE, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/presolpairwise",
         "should pairwise constraint comparison be performed in presolving?",
         &conshdlrdata->presolpairwise, TRUE, DEFAULT_PRESOLPAIRWISE, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/disjunctive", "extract disjunctive constraints?",
         &conshdlrdata->disjunctive, FALSE, DEFAULT_DISJUNCTIVE, NULL, NULL) );

   SCIP_CALL( SCIPaddLongintParam(scip,
         "constraints/" CONSHDLR_NAME "/maxnodes",
         "number of branch-and-bound nodes to solve an independent cumulative constraint (-1: no limit)?",
         &conshdlrdata->maxnodes, FALSE, DEFAULT_MAXNODES, -1LL, SCIP_LONGINT_MAX, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/detectdisjunctive", "search for conflict set via maximal cliques to detect disjunctive constraints",
         &conshdlrdata->detectdisjunctive, FALSE, DEFAULT_DETECTDISJUNCTIVE, NULL, NULL) );
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/detectvarbounds", "search for conflict set via maximal cliques to detect variable bound constraints",
         &conshdlrdata->detectvarbounds, FALSE, DEFAULT_DETECTVARBOUNDS, NULL, NULL) );

   /* conflict analysis parameters */
   SCIP_CALL( SCIPaddBoolParam(scip,
         "constraints/" CONSHDLR_NAME "/usebdwidening", "should bound widening be used during the conflict analysis?",
         &conshdlrdata->usebdwidening, FALSE, DEFAULT_USEBDWIDENING, NULL, NULL) );

   return SCIP_OKAY;
}

/** creates and captures a cumulative constraint */
SCIP_RETCODE SCIPcreateConsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           cons,               /**< pointer to hold the created constraint */
   const char*           name,               /**< name of constraint */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   SCIP_Bool             initial,            /**< should the LP relaxation of constraint be in the initial LP?
                                              *   Usually set to TRUE. Set to FALSE for 'lazy constraints'. */
   SCIP_Bool             separate,           /**< should the constraint be separated during LP processing?
                                              *   Usually set to TRUE. */
   SCIP_Bool             enforce,            /**< should the constraint be enforced during node processing?
                                              *   TRUE for model constraints, FALSE for additional, redundant constraints. */
   SCIP_Bool             check,              /**< should the constraint be checked for feasibility?
                                              *   TRUE for model constraints, FALSE for additional, redundant constraints. */
   SCIP_Bool             propagate,          /**< should the constraint be propagated during node processing?
                                              *   Usually set to TRUE. */
   SCIP_Bool             local,              /**< is constraint only valid locally?
                                              *   Usually set to FALSE. Has to be set to TRUE, e.g., for branching constraints. */
   SCIP_Bool             modifiable,         /**< is constraint modifiable (subject to column generation)?
                                              *   Usually set to FALSE. In column generation applications, set to TRUE if pricing
                                              *   adds coefficients to this constraint. */
   SCIP_Bool             dynamic,            /**< is constraint subject to aging?
                                              *   Usually set to FALSE. Set to TRUE for own cuts which
                                              *   are seperated as constraints. */
   SCIP_Bool             removable,          /**< should the relaxation be removed from the LP due to aging or cleanup?
                                              *   Usually set to FALSE. Set to TRUE for 'lazy constraints' and 'user cuts'. */
   SCIP_Bool             stickingatnode      /**< should the constraint always be kept at the node where it was added, even
                                              *   if it may be moved to a more global node?
                                              *   Usually set to FALSE. Set to TRUE to for constraints that represent node data. */
   )
{
   SCIP_CONSHDLR* conshdlr;
   SCIP_CONSDATA* consdata;

   assert(scip != NULL);

   /* find the cumulative constraint handler */
   conshdlr = SCIPfindConshdlr(scip, CONSHDLR_NAME);
   if( conshdlr == NULL )
   {
      SCIPerrorMessage("" CONSHDLR_NAME " constraint handler not found\n");
      return SCIP_PLUGINNOTFOUND;
   }

   SCIPdebugMessage("create cumulative constraint <%s> with %d jobs\n", name, nvars);

   /* create constraint data */
   SCIP_CALL( consdataCreate(scip, &consdata, vars, NULL, durations, demands, nvars, capacity, 0, INT_MAX, check) );

   /* create constraint */
   SCIP_CALL( SCIPcreateCons(scip, cons, name, conshdlr, consdata,
         initial, separate, enforce, check, propagate,
         local, modifiable, dynamic, removable, stickingatnode) );


   if( SCIPgetStage(scip) != SCIP_STAGE_PROBLEM )
   {
      SCIP_CONSHDLRDATA* conshdlrdata;

      /* get event handler */
      conshdlrdata = SCIPconshdlrGetData(conshdlr);
      assert(conshdlrdata != NULL);
      assert(conshdlrdata->eventhdlr != NULL);

      /* catch bound change events of variables */
      SCIP_CALL( consdataCatchEvents(scip, consdata, conshdlrdata->eventhdlr) );
   }

   return SCIP_OKAY;
}

/** creates and captures a cumulative constraint
 *  in its most basic version, i. e., all constraint flags are set to their basic value as explained for the
 *  method SCIPcreateConsCumulative(); all flags can be set via SCIPsetConsFLAGNAME-methods in scip.h
 *
 *  @see SCIPcreateConsCumulative() for information about the basic constraint flag configuration
 *
 *  @note the constraint gets captured, hence at one point you have to release it using the method SCIPreleaseCons()
 */
SCIP_RETCODE SCIPcreateConsBasicCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS**           cons,               /**< pointer to hold the created constraint */
   const char*           name,               /**< name of constraint */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity            /**< available cumulative capacity */
   )
{
   assert(scip != NULL);

   SCIP_CALL( SCIPcreateConsCumulative(scip, cons, name, nvars, vars, durations, demands, capacity,
         TRUE, TRUE, TRUE, TRUE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE) );

   return SCIP_OKAY;
}

/** set the left bound of the time axis to be considered (including hmin) */
SCIP_RETCODE SCIPsetHminCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint data */
   int                   hmin                /**< left bound of time axis to be considered */
   )
{
   SCIP_CONSDATA* consdata;
   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      return SCIP_INVALIDCALL;
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(hmin >= 0);
   assert(hmin <= consdata->hmax);

   consdata->hmin = hmin;

   return SCIP_OKAY;
}

/** returns the left bound of the time axis to be considered */
int SCIPgetHminCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint */
   )
{
   SCIP_CONSDATA* consdata;
   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return 0;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->hmin;
}

/** set the right bound of the time axis to be considered (not including hmax) */
SCIP_RETCODE SCIPsetHmaxCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons,               /**< constraint data */
   int                   hmax                /**< right bound of time axis to be considered */
   )
{
   SCIP_CONSDATA* consdata;
   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return SCIP_INVALIDCALL; /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);
   assert(hmax >= consdata->hmin);

   consdata->hmax = hmax;

   return SCIP_OKAY;
}

/** returns the right bound of the time axis to be considered */
int SCIPgetHmaxCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint */
   )
{
   SCIP_CONSDATA* consdata;
   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return 0;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->hmax;
}

/** returns the activities of the cumulative constraint */
SCIP_VAR** SCIPgetVarsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint data */
   )
{
   SCIP_CONSDATA* consdata;

   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return NULL;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->vars;
}

/** returns the activities of the cumulative constraint */
int SCIPgetNVarsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint data */
   )
{
   SCIP_CONSDATA* consdata;

   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return -1;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->nvars;
}

/** returns the capacity of the cumulative constraint */
int SCIPgetCapacityCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint data */
   )
{
   SCIP_CONSDATA* consdata;

   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return -1;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->capacity;
}

/** returns the durations of the cumulative constraint */
int* SCIPgetDurationsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint data */
   )
{
   SCIP_CONSDATA* consdata;

   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return NULL;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->durations;
}

/** returns the demands of the cumulative constraint */
int* SCIPgetDemandsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< constraint data */
   )
{
   SCIP_CONSDATA* consdata;

   if( strcmp(SCIPconshdlrGetName(SCIPconsGetHdlr(cons)), CONSHDLR_NAME) != 0 )
   {
      SCIPerrorMessage("constraint is not a cumulative constraint\n");
      SCIPABORT();
      return NULL;  /*lint !e527*/
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   return consdata->demands;
}

/** check for the given starting time variables with their demands and durations if the cumulative conditions for the
 *  given solution is satisfied
 */
SCIP_RETCODE SCIPcheckCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_SOL*             sol,                /**< primal solution, or NULL for current LP/pseudo solution */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            violated,           /**< pointer to store if the cumulative condition is violated */
   SCIP_CONS*            cons,               /**< constraint which is checked */
   SCIP_Bool             printreason         /**< should the reason for the violation be printed? */
   )
{
   assert(scip != NULL);
   assert(violated != NULL);

   SCIP_CALL( checkCumulativeCondition(scip, sol, nvars, vars, durations, demands, capacity, hmin, hmax,
         violated, cons, printreason) );

   return SCIP_OKAY;
}

/** normalize cumulative condition */
SCIP_RETCODE SCIPnormalizeCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int*                  capacity,           /**< pointer to store the changed cumulative capacity */
   int*                  nchgcoefs,          /**< pointer to count total number of changed coefficients */
   int*                  nchgsides           /**< pointer to count number of side changes */
   )
{
   SCIP_CALL( normalizeCumulativeCondition(scip, nvars, vars, durations, demands, capacity,
         nchgcoefs, nchgsides) );

   return SCIP_OKAY;
}

/** searches for a time point within the cumulative condition were the cumulative condition can be split */
SCIP_RETCODE SCIPsplitCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int*                  hmin,               /**< pointer to store the left bound of the effective horizon */
   int*                  hmax,               /**< pointer to store the right bound of the effective horizon */
   int*                  split               /**< point were the cumulative condition can be split */
   )
{
   SCIP_CALL( computeEffectiveHorizonCumulativeCondition(scip, nvars, vars, durations, demands, capacity,
         hmin, hmax, split) );

   return SCIP_OKAY;
}

/** presolve cumulative condition w.r.t. effective horizon by detecting irrelevant variables */
SCIP_RETCODE SCIPpresolveCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int                   hmin,               /**< left bound of time axis to be considered */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Bool*            downlocks,          /**< array storing if the variable has a down lock, or NULL */
   SCIP_Bool*            uplocks,            /**< array storing if the variable has an up lock, or NULL */
   SCIP_CONS*            cons,               /**< constraint which gets propagated, or NULL */
   SCIP_Bool*            irrelevants,        /**< array mark those variables which are irrelevant for the cumulative condition */
   int*                  nfixedvars,         /**< pointer to store the number of fixed variables */
   int*                  nchgsides,          /**< pointer to store the number of changed sides */
   SCIP_Bool*            cutoff              /**< buffer to store whether a cutoff is detected */
   )
{
   if( nvars <= 1 )
      return SCIP_OKAY;

   /* presolve constraint form the earlier start time point of view */
   SCIP_CALL( presolveConsEst(scip, nvars, vars, durations, hmin, hmax, downlocks, uplocks, cons,
         irrelevants, nfixedvars, nchgsides, cutoff) );

   /* presolve constraint form the latest completion time point of view */
   SCIP_CALL( presolveConsLct(scip, nvars, vars, durations, hmin, hmax, downlocks, uplocks, cons,
         irrelevants, nfixedvars, nchgsides, cutoff) );

   return SCIP_OKAY;
}

/** propagate the given cumulative condition */
SCIP_RETCODE SCIPpropCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_PRESOLTIMING     presoltiming,       /**< current presolving timing */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands,            /**< array containing corresponding demands */
   int                   capacity,           /**< available cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_CONS*            cons,               /**< constraint which gets propagated */
   int*                  nchgbds,            /**< pointer to store the number of variable bound changes */
   SCIP_Bool*            initialized,        /**< was conflict analysis initialized */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_Bool*            cutoff              /**< pointer to store if the cumulative condition is violated */
   )
{
   SCIP_CONSHDLR* conshdlr;
   SCIP_CONSHDLRDATA* conshdlrdata;
   SCIP_Bool redundant;

   assert(scip != NULL);
   assert(cons != NULL);
   assert(initialized != NULL);
   assert(*initialized == FALSE);
   assert(cutoff != NULL);
   assert(*cutoff == FALSE);

   /* find the cumulative constraint handler */
   conshdlr = SCIPfindConshdlr(scip, CONSHDLR_NAME);
   if( conshdlr == NULL )
   {
      SCIPerrorMessage("" CONSHDLR_NAME " constraint handler not found\n");
      return SCIP_PLUGINNOTFOUND;
   }

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   redundant = FALSE;

   SCIP_CALL( propagateCumulativeCondition(scip, conshdlrdata, presoltiming,
         nvars, vars, durations, demands, capacity,  hmin, hmax, cons,
         nchgbds, &redundant, initialized, explanation, cutoff) );

   return SCIP_OKAY;
}

/** resolve propagation w.r.t. the cumulative condition */
SCIP_RETCODE SCIPrespropCumulativeCondition(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   nvars,              /**< number of start time variables (activities) */
   SCIP_VAR**            vars,               /**< array of start time variables */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_VAR*             infervar,           /**< the conflict variable whose bound change has to be resolved */
   int                   inferinfo,          /**< the user information */
   SCIP_BOUNDTYPE        boundtype,          /**< the type of the changed bound (lower or upper bound) */
   SCIP_BDCHGIDX*        bdchgidx,           /**< the index of the bound change, representing the point of time where the change took place */
   SCIP_Real             relaxedbd,          /**< the relaxed bound which is sufficient to be explained */
   SCIP_Bool*            explanation,        /**< bool array which marks the variable which are part of the explanation if a cutoff was detected, or NULL */
   SCIP_RESULT*          result              /**< pointer to store the result of the propagation conflict resolving call */
   )
{
   SCIP_CALL( respropCumulativeCondition(scip, nvars, vars, durations, demands, capacity, hmin, hmax,
         infervar, intToInferInfo(inferinfo), boundtype, bdchgidx, relaxedbd, TRUE, explanation, result) );

   return SCIP_OKAY;
}

/** this method visualizes the cumulative structure in GML format */
SCIP_RETCODE SCIPvisualizeConsCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_CONS*            cons                /**< cumulative constraint */
   )
{
   SCIP_CONSDATA* consdata;
   SCIP_HASHTABLE* vars;
   FILE* file;
   SCIP_VAR* var;
   char filename[SCIP_MAXSTRLEN];
   int nvars;
   int v;

   /* open file */
   (void)SCIPsnprintf(filename, SCIP_MAXSTRLEN, "%s.gml", SCIPconsGetName(cons));
   file = fopen(filename, "w");

   /* check if the file was open */
   if( file == NULL )
   {
      SCIPerrorMessage("cannot create file <%s> for writing\n", filename);
      SCIPprintSysError(filename);
      return SCIP_FILECREATEERROR;
   }

   consdata = SCIPconsGetData(cons);
   assert(consdata != NULL);

   nvars = consdata->nvars;

   SCIP_CALL( SCIPhashtableCreate(&vars, SCIPblkmem(scip), SCIPcalcHashtableSize(nvars),
         SCIPvarGetHashkey, SCIPvarIsHashkeyEq, SCIPvarGetHashkeyVal, NULL) );

   /* create opening of the GML format */
   SCIPgmlWriteOpening(file,  TRUE);

   for( v = 0; v < nvars; ++v )
   {
      char color[SCIP_MAXSTRLEN];

      var = consdata->vars[v];
      assert(var != NULL);

      SCIP_CALL( SCIPhashtableInsert(vars, (void*)var) );

      if( SCIPvarGetUbGlobal(var) - SCIPvarGetLbGlobal(var) < 0.5 )
         (void)SCIPsnprintf(color, SCIP_MAXSTRLEN, "%s", "#0000ff");
      else if( !consdata->downlocks[v] || !consdata->uplocks[v] )
         (void)SCIPsnprintf(color, SCIP_MAXSTRLEN, "%s", "#00ff00");
      else
         (void)SCIPsnprintf(color, SCIP_MAXSTRLEN, "%s", "#ff0000");

      SCIPgmlWriteNode(file, (unsigned int)(size_t)var, SCIPvarGetName(var), "rectangle", color, NULL);
   }

   for( v = 0; v < nvars; ++v )
   {
      SCIP_VAR** vbdvars;
      int nvbdvars;
      int b;

      var = consdata->vars[v];
      assert(var != NULL);

      vbdvars = SCIPvarGetVlbVars(var);
      nvbdvars = SCIPvarGetNVlbs(var);

      for( b = 0; b < nvbdvars; ++b )
      {
         if( SCIPhashtableExists(vars, (void*)vbdvars[b]) )
         {
            SCIPgmlWriteArc(file, (unsigned int)(size_t)vbdvars[b], (unsigned int)(size_t)var, NULL, NULL);
         }
      }

#if 0
      vbdvars = SCIPvarGetVubVars(var);
      nvbdvars = SCIPvarGetNVubs(var);

      for( b = 0; b < nvbdvars; ++b )
      {
         if( SCIPhashtableExists(vars, vbdvars[b]) )
         {
            SCIPgmlWriteArc(file, (unsigned int)(size_t)var, (unsigned int)(size_t)vbdvars[b], NULL, NULL);
         }
      }
#endif
   }

   /* create closing of the GML format */
   SCIPgmlWriteClosing(file);

   /* close file */
   fclose(file);

   SCIPhashtableFree(&vars);

   return SCIP_OKAY;
}

/** sets method to solve an individual cumulative condition */
SCIP_RETCODE SCIPsetSolveCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_DECL_SOLVECUMULATIVE((*solveCumulative)) /**< method to use an individual cumulative condition */
   )
{
   SCIP_CONSHDLR* conshdlr;
   SCIP_CONSHDLRDATA* conshdlrdata;

   /* find the cumulative constraint handler */
   conshdlr = SCIPfindConshdlr(scip, CONSHDLR_NAME);
   if( conshdlr == NULL )
   {
      SCIPerrorMessage("" CONSHDLR_NAME " constraint handler not found\n");
      return SCIP_PLUGINNOTFOUND;
   }

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   conshdlrdata->solveCumulative = solveCumulative;

   return SCIP_OKAY;
}

/** solves given cumulative condition as independent sub problem
 *
 *  @note If the problem was solved to the earliest start times (ests) and latest start times (lsts) array contain the
 *        solution values; If the problem was not solved these two arrays contain the global bounds at the time the sub
 *        solver was interrupted.
 */
SCIP_RETCODE SCIPsolveCumulative(
   SCIP*                 scip,               /**< SCIP data structure */
   int                   njobs,              /**< number of jobs (activities) */
   SCIP_Real*            ests,               /**< array with the earlier start time for each job */
   SCIP_Real*            lsts,               /**< array with the latest start time for each job */
   SCIP_Real*            objvals,            /**< array of objective coefficients for each job (linear objective function), or NULL if none */
   int*                  durations,          /**< array of durations */
   int*                  demands,            /**< array of demands */
   int                   capacity,           /**< cumulative capacity */
   int                   hmin,               /**< left bound of time axis to be considered (including hmin) */
   int                   hmax,               /**< right bound of time axis to be considered (not including hmax) */
   SCIP_Real             timelimit,          /**< time limit for solving in seconds */
   SCIP_Real             memorylimit,        /**< memory limit for solving in mega bytes (MB) */
   SCIP_Longint          maxnodes,           /**< maximum number of branch-and-bound nodes to solve the single cumulative constraint  (-1: no limit) */
   SCIP_Bool*            solved,             /**< pointer to store if the problem is solved (to optimality) */
   SCIP_Bool*            infeasible,         /**< pointer to store if the problem is infeasible */
   SCIP_Bool*            unbounded,          /**< pointer to store if the problem is unbounded */
   SCIP_Bool*            error               /**< pointer to store if an error occurred */
   )
{
   SCIP_CONSHDLR* conshdlr;
   SCIP_CONSHDLRDATA* conshdlrdata;

   (*solved) = TRUE;
   (*infeasible) = FALSE;
   (*unbounded) = FALSE;
   (*error) = FALSE;

   if( njobs == 0 )
      return SCIP_OKAY;

   /* find the cumulative constraint handler */
   conshdlr = SCIPfindConshdlr(scip, CONSHDLR_NAME);
   if( conshdlr == NULL )
   {
      SCIPerrorMessage("" CONSHDLR_NAME " constraint handler not found\n");
      (*error) = TRUE;
      return SCIP_PLUGINNOTFOUND;
   }

   conshdlrdata = SCIPconshdlrGetData(conshdlr);
   assert(conshdlrdata != NULL);

   /* abort if no time is left or not enough memory to create a copy of SCIP, including external memory usage */
   if( timelimit > 0.0 && memorylimit > 10 )
   {
      SCIP_CALL( conshdlrdata->solveCumulative(njobs, ests, lsts, objvals, durations, demands, capacity,
            hmin, hmax, timelimit, memorylimit, maxnodes, solved, infeasible, unbounded, error) );
   }

   return SCIP_OKAY;
}

/** creates the worst case resource profile, that is, all jobs are inserted with the earliest start and latest
 *  completion time
 */
SCIP_RETCODE SCIPcreateWorstCaseProfile(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_PROFILE*         profile,            /**< resource profile */
   int                   nvars,              /**< number of variables (jobs) */
   SCIP_VAR**            vars,               /**< array of integer variable which corresponds to starting times for a job */
   int*                  durations,          /**< array containing corresponding durations */
   int*                  demands             /**< array containing corresponding demands */
   )
{
   SCIP_VAR* var;
   SCIP_HASHMAP* addedvars;
   int* copydemands;
   int* perm;
   int duration;
   int impliedest;
   int est;
   int impliedlct;
   int lct;
   int v;

   /* create hash map for variables which are added, mapping to their duration */
   SCIP_CALL( SCIPhashmapCreate(&addedvars, SCIPblkmem(scip), SCIPcalcHashtableSize(nvars)) );

   SCIP_CALL( SCIPallocBufferArray(scip, &perm, nvars) );
   SCIP_CALL( SCIPallocBufferArray(scip, &copydemands, nvars) );

   /* sort variables  w.r.t. job demands */
   for( v = 0; v < nvars; ++v )
   {
      copydemands[v] = demands[v];
      perm[v] = v;
   }
   SCIPsortDownIntInt(copydemands, perm, nvars);

   /* add each job with its earliest start and latest completion time into the resource profile */
   for( v = 0; v < nvars; ++v )
   {
      int idx;

      idx = perm[v];
      assert(idx >= 0 && idx < nvars);

      var = vars[idx];
      assert(var != NULL);

      duration = durations[idx];
      assert(duration > 0);

      est = SCIPconvertRealToInt(scip, SCIPvarGetLbLocal(var));
      SCIP_CALL( computeImpliedEst(scip, var, addedvars, &impliedest) );

      lct = SCIPconvertRealToInt(scip, SCIPvarGetUbLocal(var)) + duration;
      SCIP_CALL( computeImpliedLct(scip, var, duration, addedvars, &impliedlct) );

      if( impliedest < impliedlct )
      {
         SCIP_Bool infeasible;
         int pos;

         SCIP_CALL( SCIPprofileInsertCore(profile, impliedest, impliedlct, copydemands[v], &pos, &infeasible) );
         assert(!infeasible);
         assert(pos == -1);
      }

      if( est == impliedest && lct == impliedlct )
      {
         SCIP_CALL( SCIPhashmapInsert(addedvars, (void*)var, (void*)(size_t)duration) );
      }
   }

   SCIPfreeBufferArray(scip, &copydemands);
   SCIPfreeBufferArray(scip, &perm);

   SCIPhashmapFree(&addedvars);

   return SCIP_OKAY;
}

/** computes w.r.t. the given worst case resource profile the first time point where the given capacity can be violated */
int SCIPcomputeHmin(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_PROFILE*         profile,            /**< worst case resource profile */
   int                   capacity            /**< capacity to check */
   )
{
   int* timepoints;
   int* loads;
   int ntimepoints;
   int t;

   ntimepoints = SCIPprofileGetNTimepoints(profile);
   timepoints = SCIPprofileGetTimepoints(profile);
   loads = SCIPprofileGetLoads(profile);

   /* find first time point which potentially violates the capacity restriction */
   for( t = 0; t < ntimepoints - 1; ++t )
   {
      /* check if the time point exceed w.r.t. worst case profile the capacity */
      if( loads[t] > capacity )
      {
         assert(t == 0 || loads[t-1] <= capacity);
         return timepoints[t];
      }
   }

   return INT_MAX;
}

/** computes w.r.t. the given worst case resource profile the first time point where the given capacity is satisfied for sure */
int SCIPcomputeHmax(
   SCIP*                 scip,               /**< SCIP data structure */
   SCIP_PROFILE*         profile,            /**< worst case profile */
   int                   capacity            /**< capacity to check */
   )
{
   int* timepoints;
   int* loads;
   int ntimepoints;
   int t;

   ntimepoints = SCIPprofileGetNTimepoints(profile);
   timepoints = SCIPprofileGetTimepoints(profile);
   loads = SCIPprofileGetLoads(profile);

   /* find last time point which potentially violates the capacity restriction */
   for( t = ntimepoints - 1; t >= 0; --t )
   {
      /* check if at time point t the worst case resource profile  exceeds the capacity */
      if( loads[t] > capacity )
      {
         assert(t == ntimepoints-1 || loads[t+1] <= capacity);
         return timepoints[t+1];
      }
   }

   return INT_MIN;
}

/**@} */