GomoryHuTree.cpp

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/**@file   GomoryHuTree.cpp
 * @brief  generator for global cuts in undirected graphs
 * @author Georg Skorobohatyj
 * @author Timo Berthold
 */

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


#include <stdio.h>
#include <assert.h>

#include "objscip/objscip.h"
#include "GomoryHuTree.h"

/** epsilon value for numerical comparisons */
#define  EPS  1.0E-10

/* static variables */
static GRAPHNODE** active;
static long* number;
static long max_dist;
static long bound;
static SCIP_Bool co_check;


/** create a graph */
SCIP_Bool create_graph(
   int                   n,                  /**< number of nodes */
   int                   m,                  /**< number of edges */
   GRAPH**               gr                  /**< pointer to store graph */
   )
{
   assert( gr != NULL );

   BMSallocMemory(gr);
   if( *gr == NULL )
      return FALSE;

   BMSallocMemoryArray(&(*gr)->nodes, n);
   if( (*gr)->nodes == NULL )
   {
      BMSfreeMemory(gr);
      return FALSE;
   }

   BMSallocMemoryArray(&(*gr)->edges, m);
   if( (*gr)->edges == NULL )
   {
      BMSfreeMemoryArray(&(*gr)->nodes);
      BMSfreeMemory(gr);
      return FALSE;
   }
   (*gr)->nuses = 1;
   (*gr)->nnodes = n;
   (*gr)->nedges = m/2;
   (*gr)->nedgesnonzero = m/2;

   return TRUE;
}

/** free a graph */
static
void free_graph(
   GRAPH**               gr                  /**< pointer a graph */
   )
{
   assert(gr != NULL);
   assert(*gr != NULL);
   assert((*gr)->nuses == 0);

   BMSfreeMemory(&(*gr)->nodes);
   BMSfreeMemory(&(*gr)->edges);
   BMSfreeMemory(gr);
}

/** capture graph */
void capture_graph(
   GRAPH*                gr                  /**< graph */
   )
{
   assert(gr != NULL);

   ++gr->nuses;
}

/** release graph */
void release_graph(
   GRAPH**               gr                  /**< graph */
   )
{
   assert(gr != NULL);
   assert(*gr != NULL);

   --(*gr)->nuses;

   if( (*gr)->nuses == 0 )
      free_graph(gr);
   *gr = NULL;
}

/** initialize maximum flow computation */
static
SCIP_Bool init_maxflow(
   long                  n                   /**< number of nodes */
   )
{
   active = (GRAPHNODE**) malloc((n+1L) * sizeof (GRAPHNODE*));

   /* holds stacks of active nodes arranged by distances */
   if ( active == (GRAPHNODE **) 0 )
   {
      printf ("Unable to allocate memory\n");
      return FALSE;
   }

   number = (long *) malloc ((n+1L) * sizeof (long));

   /* counts occurences of node distances in set of alive nodes, i.e., nodes not contained in the set of nodes
    * disconnected from the sink */
   if ( number == (long *) 0 )
   {
      printf ("Unable to allocate memory\n");
      return FALSE;
   }
   co_check = TRUE;

   return TRUE;
}

/** free initialization data structures */
static
void fini_maxflow(void)
{
   free(active);
   free(number);
}

/** global relabel operation */
static
void global_relabel(
   GRAPH*                gr,                 /**< graph */
   GRAPHNODE*            tptr                /**< node to be relabeled */
   )
{
   /* breadth first search to get exact distance labels from sink with reordering of stack of active nodes */
   GRAPHNODE* front;
   GRAPHNODE* rear;
   GRAPHNODE* nptr;
   GRAPHNODE** ptr;
   GRAPHEDGE* eptr;
   long n;
   long level;
   long count;
   long i;

   n = gr->nnodes;
   for ( nptr = &(gr->nodes[n-1L]); nptr >= gr->nodes; nptr-- )
   {
      nptr->unmarked = TRUE;
      nptr->stack_link = NULL;
      nptr->scan_ptr = nptr->first_edge;
      while( nptr->scan_ptr != NULL && nptr->scan_ptr->cap <= EPS )
         nptr->scan_ptr = nptr->scan_ptr->next;
   }
   tptr->unmarked = FALSE;

   /* initialize stack of active nodes */
   for ( ptr = &(active[n]); ptr >= active; ptr-- )
      *ptr = NULL;

   for ( i = 0L; i <= n; i++ )
      number[i] = 0L;

   max_dist = 0L;
   count = 1L;     /* number of alive nodes */
   front = tptr;
   rear = front;

 bfs_next:
   level = rear->dist + 1L;
   eptr = rear->first_edge;
   while ( eptr != NULL )
   {
      nptr = eptr->adjac;
      if ( nptr->alive && nptr->unmarked && eptr->back->rcap > EPS )
      {
         assert(eptr->cap > EPS);
         assert(eptr->back->cap > EPS);

         nptr->unmarked = FALSE;
         nptr->dist = (int) level;
         ++count;
         ++number[level];

         if ( nptr->excess > EPS )
         {
            nptr->stack_link = active[level];
            active[level] = nptr;
            max_dist = level;
         }
         front->bfs_link = nptr;
         front = nptr;
      }
      eptr = eptr->next;
   }

   if ( front == rear )
      goto bfs_ready;

   rear = rear->bfs_link;
   goto bfs_next;

 bfs_ready:

   if ( count < bound )
   {
      /* identify nodes that are marked alive but have not been reached by BFS and mark them as dead */
      for ( nptr = &(gr->nodes[n-1L]); nptr >= gr->nodes; nptr-- )
      {
         if ( nptr->unmarked && nptr->alive )
         {
            nptr->dist = (int) n;
            nptr->alive = FALSE;
         }
      }
      bound = count;
   }
}

/** compute maximum flow
 *
 * Determines maximum flow and minimum cut between nodes s (= *s_ptr) and t (= *t_ptr) in an undirected graph
 *
 * References:
 * A. Goldberg/ E. Tarjan: "A New Approach to the Maximum Flow Problem", in Proc. 18th ACM Symp. on Theory of Computing, 1986.
 */
static
double maxflow(
   GRAPH*                gr,                 /**< graph */
   GRAPHNODE*            s_ptr,              /**< start node */
   GRAPHNODE*            t_ptr               /**< target node */
   )
{
   GRAPHNODE* aptr;
   GRAPHNODE* nptr;
   GRAPHNODE* q_front;
   GRAPHNODE* q_rear;
   GRAPHEDGE* eptr;
   long n;
   long m;
   long m0;
   long level;
   long i;
   long n_discharge;
   double incre;
   long dmin;
   double cap;

   /* node ids range from 1 to n, node array indices range from 0 to n-1 */
   n = gr->nnodes;
   for ( nptr = &(gr->nodes[n-1L]); nptr >= gr->nodes; nptr-- )
   {
      nptr->scan_ptr = nptr->first_edge;
      while( nptr->scan_ptr != NULL && nptr->scan_ptr->cap <= EPS )
         nptr->scan_ptr = nptr->scan_ptr->next;

      if ( nptr->scan_ptr == NULL )
      {
         fprintf(stderr, "isolated node in input graph\n");
         return FALSE;
      }

      nptr->excess = 0.0;
      nptr->stack_link = NULL;
      nptr->alive = TRUE;
      nptr->unmarked = TRUE;
   }

   m = gr->nedgesnonzero;
   m0 = gr->nedges;
   for ( eptr = &(gr->edges[m-1L]); eptr >= gr->edges; eptr-- )
      eptr->rcap = eptr->cap;

   for ( eptr = &(gr->edges[m0+m-1L]); eptr >= &(gr->edges[m0]); eptr-- )
      eptr->rcap = eptr->cap;

   for ( i = n; i >= 0L; i-- )
   {
      number[i] = 0L;
      active[i] = NULL;
   }
   t_ptr->dist = 0L;

   /* breadth first search to get exact distances from sink and for test of graph connectivity */
   t_ptr->unmarked = FALSE;
   q_front = t_ptr;
   q_rear = q_front;

 bfs_next:
   level = q_rear->dist + 1L;
   eptr = q_rear->first_edge;

   while ( eptr != NULL )
   {
      assert(eptr->back->cap == eptr->back->rcap); /*lint !e777*/
      if ( eptr->adjac->unmarked && eptr->back->rcap > EPS )
      {
         nptr = eptr->adjac;
         nptr->unmarked = FALSE;
         nptr->dist = (int) level;
         ++number[level];
         q_front->bfs_link = nptr;
         q_front = nptr;
      }
      eptr = eptr->next;
   }

   if ( q_rear == q_front )
      goto bfs_ready;

   q_rear = q_rear->bfs_link;
   goto bfs_next;

 bfs_ready:
   if ( co_check )
   {
      co_check = FALSE;
      for ( nptr = &(gr->nodes[n-1]); nptr >= gr->nodes; --nptr )
      {
         if ( nptr->unmarked )
            return (-1.0);
      }
   }

   s_ptr->dist = (int) n; /* number[0] and number[n] not required */
   t_ptr->dist = 0L;
   t_ptr->excess = 1.0;  /* to be subtracted again */

   /* initial preflow push from source node */
   max_dist = 0L;  /* = max_dist of active nodes */
   eptr = s_ptr->first_edge;

   while ( eptr != NULL )
   {
      if( eptr->cap > EPS )
      {
         nptr = eptr->adjac;
         cap = eptr->rcap;
         nptr->excess += cap;
         s_ptr->excess -= cap;
         eptr->back->rcap += cap;
         eptr->rcap = 0.0;

         if ( nptr != t_ptr && nptr->excess <= cap + EPS )
         {
            /* push node nptr onto stack for nptr->dist, but only once in case of double edges */
            nptr->stack_link = active[nptr->dist];
            active[nptr->dist] = nptr;
            if ( nptr->dist > max_dist )
               max_dist = nptr->dist;
         }
      }
      eptr = eptr->next;
   }

   s_ptr->alive = FALSE;
   bound = n;
   n_discharge = 0L;

   /* main loop */
   do
   {
      /* get maximum distance active node */
      aptr = active[max_dist];
      while ( aptr != NULL )
      {
         active[max_dist] = aptr->stack_link;
         eptr = aptr->scan_ptr;
         assert(eptr != NULL);
         assert(eptr->back != NULL);
         assert(eptr->cap > EPS);

      edge_scan:  /* for current active node  */
         assert(eptr != NULL);
         assert(eptr->back != NULL);
         assert(eptr->cap > EPS);

         nptr = eptr->adjac;
         assert(nptr != NULL);
         if ( nptr->dist == aptr->dist - 1L && eptr->rcap > EPS )
         {
            incre = aptr->excess;
            if ( incre <= eptr->rcap )
            {
               /* perform a non saturating push */
               eptr->rcap -= incre;
               eptr->back->rcap += incre;
               aptr->excess = 0.0;
               nptr->excess += incre;

               if ( nptr->excess <= incre + EPS )
               {
                  /* push nptr onto active stack */
                  nptr->stack_link = active[nptr->dist];
                  active[nptr->dist] = nptr;
               }

               assert(eptr->cap > EPS);
               aptr->scan_ptr = eptr;

               goto node_ready;
            }
            else
            {
               /* perform a saturating push */
               incre = eptr->rcap;
               eptr->back->rcap += incre;
               aptr->excess -= incre;
               nptr->excess += incre;
               eptr->rcap = 0.0;

               if ( nptr->excess <= incre + EPS )
               {
                  /* push nptr onto active stack */
                  nptr->stack_link = active[nptr->dist];
                  active[nptr->dist] = nptr;
               }

               if ( aptr->excess <= EPS )
               {
                  assert(eptr->cap > EPS);
                  aptr->scan_ptr = eptr;

                  goto node_ready;
               }
            }
         }

         /* go to the next non-zero edge */
         do
         {
            eptr = eptr->next;
         }
         while( eptr != NULL && eptr->cap <= EPS );

         if ( eptr == NULL )
         {
            /* all incident arcs scanned, but node still has positive excess, check if for all nptr
             * nptr->dist != aptr->dist */
            if ( number[aptr->dist] == 1L )
            {
               /* put all nodes v with dist[v] >= dist[a] into the set of "dead" nodes since they are disconnected from
                * the sink */
               for ( nptr = &(gr->nodes[n-1L]); nptr >= gr->nodes; nptr-- )
               {
                  if ( nptr->alive && nptr->dist > aptr->dist )
                  {
                     --number[nptr->dist];
                     active[nptr->dist] = NULL;
                     nptr->alive = FALSE;
                     nptr->dist = (int) n;
                     --bound;
                  }
               }
               --number[aptr->dist];
               active[aptr->dist] = NULL;
               aptr->alive = FALSE;
               aptr->dist = (int) n;
               --bound;

               goto node_ready;
            }
            else
            {
               /* determine new label value */
               dmin = n;
               aptr->scan_ptr = NULL;
               eptr = aptr->first_edge;
               while ( eptr != NULL )
               {
                  assert(eptr->rcap <= EPS || eptr->cap > EPS);
                  if ( eptr->adjac->dist < dmin && eptr->rcap > EPS )
                  {
                     assert(eptr->cap > EPS);
                     dmin = eptr->adjac->dist;
                     if ( aptr->scan_ptr == NULL )
                        aptr->scan_ptr = eptr;
                  }
                  eptr = eptr->next;
               }

               if ( ++dmin < bound )
               {
                  /* ordinary relabel operation */
                  --number[aptr->dist];
                  aptr->dist = (int) dmin;
                  ++number[dmin];
                  max_dist = dmin;
                  eptr = aptr->scan_ptr;
                  assert(eptr != NULL);
                  assert(eptr->cap > EPS);

                  goto edge_scan;
               }
               else
               {
                  aptr->alive = FALSE;
                  --number[aptr->dist];
                  aptr->dist = (int) n;
                  --bound;

                  goto node_ready;
               }
            }
         }
         else
            goto edge_scan;

      node_ready:
         ++n_discharge;
         if ( n_discharge == n )
         {
            n_discharge = 0L;
            global_relabel (gr, t_ptr);
         }
         aptr = active[max_dist];
      }
      --max_dist;
   }
   while ( max_dist > 0L );

   return (int) (t_ptr->excess - 1.0L);
}

/** determine whether node i is on the path to the root starting from j */
static
SCIP_Bool nodeOnRootPath(
   GRAPH*                gr,                 /**< graph */
   int                   i,                  /**< node to search for */
   int                   j                   /**< starting node */
   )
{
   while( i != j && j != 0 )
   {
      j = gr->nodes[j].parent->id;
   }

   if( i == j )
      return TRUE;
   else
      return FALSE;
}

/** constructs a list of cuts for a TSP relaxation polytope from a Gomory Hu Tree
 *
 *  If the non-zero-edges of a TSP relaxation induce a non-connected graph, an according cut is generated, using
 *  information from BFS in method maxflow.
 */
static
void constructCutList(
   GRAPH*                gr,                 /**< graph */
   SCIP_Bool**           cuts,               /**< array of arrays to store cuts */
   int*                  ncuts,              /**< pointer to store number of cuts */
   double                minviol             /**< minimal violation of a cut to be returned */
   )
{
   int k = 0;

   for( int i = 1; i < gr->nnodes; i++ )
   {
      if( gr->nodes[i].mincap < 2.0 - minviol )
      {
         cuts[k][0] = FALSE;
         for( int j = 1 ; j < gr->nnodes; j++ )
            cuts[k][j] = nodeOnRootPath(gr, i, j);
         k++;
      }
   }
   *ncuts = k;
}

/** construct a single cut */
static
void constructSingleCut(
   GRAPH*                gr,                 /**< graph */
   SCIP_Bool**           cuts                /**< array of arrays to store cuts */
   )
{
   cuts[0][0] = FALSE;
   for( int i = 1; i < gr->nnodes; i++ )
      cuts[0][i]=gr->nodes[i].unmarked;
}

/** Determines Gomory/Hu cut tree for input graph with capacitated edges
 *
 * The tree structures is represented by parent pointers which are part of the node structure, the capacity of a tree
 * edge is stored at the child node, the root of the cut tree is the first node in the list of graph nodes
 * (&gr->nodes[0]). The implementation is described in [1].
 *
 * References:
 * 1) D. Gusfield: "Very Simple Algorithms and Programs for  All Pairs Network Flow Analysis",
 *    Computer Science Division, University of California, Davis, 1987.
 *
 * 2) R.E. Gomory and T.C. Hu: "Multi-Terminal Network Flows",
 *    SIAM J. Applied Math. 9 (1961), 551-570.
 */
SCIP_Bool ghc_tree(
   GRAPH*                gr,                 /**< graph */
   SCIP_Bool**           cuts,               /**< array of arrays to store cuts */
   int*                  ncuts,              /**< pointer to store number of cuts */
   double                minviol             /**< minimal violation of a cut to be returned */
   )
{
   GRAPHNODE* nptr;
   GRAPHNODE* nptr1;
   GRAPHNODE* nptrn;
   GRAPHNODE* sptr;
   GRAPHNODE* tptr;
   double maxfl;
   long n;

   n = gr->nnodes;

   if ( ! init_maxflow(n) )
      return FALSE;

   nptr1 = gr->nodes;
   nptrn = &(gr->nodes[n-1L]);
   for ( nptr = nptrn; nptr >= nptr1; nptr-- )
      nptr->parent = nptr1;

   for ( sptr = &(gr->nodes[1L]); sptr <= nptrn; sptr++ )
   {
      tptr = sptr->parent;
      maxfl = maxflow(gr, sptr, tptr);

      /* maxfl < 0 <=> graph is not connected => generate cut */
      if ( maxfl < 0L )
      {
         constructSingleCut(gr,cuts);
         *ncuts = 1;
         fini_maxflow();
         return TRUE;
      }

      sptr->mincap = maxfl;
      for ( nptr = &(gr->nodes[1L]); nptr <= nptrn; nptr++ )
      {
         if ( nptr != sptr && ! nptr->alive && nptr->parent == tptr )
            nptr->parent = sptr;
      }

      if ( ! tptr->parent->alive )
      {
         sptr->parent = tptr->parent;
         tptr->parent = sptr;
         sptr->mincap = tptr->mincap;
         tptr->mincap = maxfl;
      }
   }
   fini_maxflow();
   constructCutList(gr, cuts, ncuts, minviol);

   return TRUE;
}