COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 832:9ae88e7c04a7

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
22/// \ingroup min_cost_flow_algs
23///
24/// \file
25/// \brief Network Simplex algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
31#include <lemon/core.h>
32#include <lemon/math.h>
33
34namespace lemon {
35
36  /// \addtogroup min_cost_flow_algs
37  /// @{
38
39  /// \brief Implementation of the primal Network Simplex algorithm
40  /// for finding a \ref min_cost_flow "minimum cost flow".
41  ///
42  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
43  /// for finding a \ref min_cost_flow "minimum cost flow".
44  /// This algorithm is a specialized version of the linear programming
45  /// simplex method directly for the minimum cost flow problem.
46  /// It is one of the most efficient solution methods.
47  ///
48  /// In general this class is the fastest implementation available
49  /// in LEMON for the minimum cost flow problem.
50  /// Moreover it supports both directions of the supply/demand inequality
51  /// constraints. For more information see \ref SupplyType.
52  ///
53  /// Most of the parameters of the problem (except for the digraph)
54  /// can be given using separate functions, and the algorithm can be
55  /// executed using the \ref run() function. If some parameters are not
56  /// specified, then default values will be used.
57  ///
58  /// \tparam GR The digraph type the algorithm runs on.
59  /// \tparam V The value type used for flow amounts, capacity bounds
60  /// and supply values in the algorithm. By default it is \c int.
61  /// \tparam C The value type used for costs and potentials in the
62  /// algorithm. By default it is the same as \c V.
63  ///
64  /// \warning Both value types must be signed and all input data must
65  /// be integer.
66  ///
67  /// \note %NetworkSimplex provides five different pivot rule
68  /// implementations, from which the most efficient one is used
69  /// by default. For more information see \ref PivotRule.
70  template <typename GR, typename V = int, typename C = V>
71  class NetworkSimplex
72  {
73  public:
74
75    /// The type of the flow amounts, capacity bounds and supply values
76    typedef V Value;
77    /// The type of the arc costs
78    typedef C Cost;
79
80  public:
81
82    /// \brief Problem type constants for the \c run() function.
83    ///
84    /// Enum type containing the problem type constants that can be
85    /// returned by the \ref run() function of the algorithm.
86    enum ProblemType {
87      /// The problem has no feasible solution (flow).
88      INFEASIBLE,
89      /// The problem has optimal solution (i.e. it is feasible and
90      /// bounded), and the algorithm has found optimal flow and node
91      /// potentials (primal and dual solutions).
92      OPTIMAL,
93      /// The objective function of the problem is unbounded, i.e.
94      /// there is a directed cycle having negative total cost and
95      /// infinite upper bound.
96      UNBOUNDED
97    };
98   
99    /// \brief Constants for selecting the type of the supply constraints.
100    ///
101    /// Enum type containing constants for selecting the supply type,
102    /// i.e. the direction of the inequalities in the supply/demand
103    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
104    ///
105    /// The default supply type is \c GEQ, the \c LEQ type can be
106    /// selected using \ref supplyType().
107    /// The equality form is a special case of both supply types.
108    enum SupplyType {
109      /// This option means that there are <em>"greater or equal"</em>
110      /// supply/demand constraints in the definition of the problem.
111      GEQ,
112      /// This option means that there are <em>"less or equal"</em>
113      /// supply/demand constraints in the definition of the problem.
114      LEQ
115    };
116   
117    /// \brief Constants for selecting the pivot rule.
118    ///
119    /// Enum type containing constants for selecting the pivot rule for
120    /// the \ref run() function.
121    ///
122    /// \ref NetworkSimplex provides five different pivot rule
123    /// implementations that significantly affect the running time
124    /// of the algorithm.
125    /// By default \ref BLOCK_SEARCH "Block Search" is used, which
126    /// proved to be the most efficient and the most robust on various
127    /// test inputs according to our benchmark tests.
128    /// However another pivot rule can be selected using the \ref run()
129    /// function with the proper parameter.
130    enum PivotRule {
131
132      /// The First Eligible pivot rule.
133      /// The next eligible arc is selected in a wraparound fashion
134      /// in every iteration.
135      FIRST_ELIGIBLE,
136
137      /// The Best Eligible pivot rule.
138      /// The best eligible arc is selected in every iteration.
139      BEST_ELIGIBLE,
140
141      /// The Block Search pivot rule.
142      /// A specified number of arcs are examined in every iteration
143      /// in a wraparound fashion and the best eligible arc is selected
144      /// from this block.
145      BLOCK_SEARCH,
146
147      /// The Candidate List pivot rule.
148      /// In a major iteration a candidate list is built from eligible arcs
149      /// in a wraparound fashion and in the following minor iterations
150      /// the best eligible arc is selected from this list.
151      CANDIDATE_LIST,
152
153      /// The Altering Candidate List pivot rule.
154      /// It is a modified version of the Candidate List method.
155      /// It keeps only the several best eligible arcs from the former
156      /// candidate list and extends this list in every iteration.
157      ALTERING_LIST
158    };
159   
160  private:
161
162    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
163
164    typedef std::vector<int> IntVector;
165    typedef std::vector<bool> BoolVector;
166    typedef std::vector<Value> ValueVector;
167    typedef std::vector<Cost> CostVector;
168
169    // State constants for arcs
170    enum ArcStateEnum {
171      STATE_UPPER = -1,
172      STATE_TREE  =  0,
173      STATE_LOWER =  1
174    };
175
176  private:
177
178    // Data related to the underlying digraph
179    const GR &_graph;
180    int _node_num;
181    int _arc_num;
182    int _all_arc_num;
183    int _search_arc_num;
184
185    // Parameters of the problem
186    bool _have_lower;
187    SupplyType _stype;
188    Value _sum_supply;
189
190    // Data structures for storing the digraph
191    IntNodeMap _node_id;
192    IntArcMap _arc_id;
193    IntVector _source;
194    IntVector _target;
195
196    // Node and arc data
197    ValueVector _lower;
198    ValueVector _upper;
199    ValueVector _cap;
200    CostVector _cost;
201    ValueVector _supply;
202    ValueVector _flow;
203    CostVector _pi;
204
205    // Data for storing the spanning tree structure
206    IntVector _parent;
207    IntVector _pred;
208    IntVector _thread;
209    IntVector _rev_thread;
210    IntVector _succ_num;
211    IntVector _last_succ;
212    IntVector _dirty_revs;
213    BoolVector _forward;
214    IntVector _state;
215    int _root;
216
217    // Temporary data used in the current pivot iteration
218    int in_arc, join, u_in, v_in, u_out, v_out;
219    int first, second, right, last;
220    int stem, par_stem, new_stem;
221    Value delta;
222
223  public:
224 
225    /// \brief Constant for infinite upper bounds (capacities).
226    ///
227    /// Constant for infinite upper bounds (capacities).
228    /// It is \c std::numeric_limits<Value>::infinity() if available,
229    /// \c std::numeric_limits<Value>::max() otherwise.
230    const Value INF;
231
232  private:
233
234    // Implementation of the First Eligible pivot rule
235    class FirstEligiblePivotRule
236    {
237    private:
238
239      // References to the NetworkSimplex class
240      const IntVector  &_source;
241      const IntVector  &_target;
242      const CostVector &_cost;
243      const IntVector  &_state;
244      const CostVector &_pi;
245      int &_in_arc;
246      int _search_arc_num;
247
248      // Pivot rule data
249      int _next_arc;
250
251    public:
252
253      // Constructor
254      FirstEligiblePivotRule(NetworkSimplex &ns) :
255        _source(ns._source), _target(ns._target),
256        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
257        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
258        _next_arc(0)
259      {}
260
261      // Find next entering arc
262      bool findEnteringArc() {
263        Cost c;
264        for (int e = _next_arc; e < _search_arc_num; ++e) {
265          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
266          if (c < 0) {
267            _in_arc = e;
268            _next_arc = e + 1;
269            return true;
270          }
271        }
272        for (int e = 0; e < _next_arc; ++e) {
273          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
274          if (c < 0) {
275            _in_arc = e;
276            _next_arc = e + 1;
277            return true;
278          }
279        }
280        return false;
281      }
282
283    }; //class FirstEligiblePivotRule
284
285
286    // Implementation of the Best Eligible pivot rule
287    class BestEligiblePivotRule
288    {
289    private:
290
291      // References to the NetworkSimplex class
292      const IntVector  &_source;
293      const IntVector  &_target;
294      const CostVector &_cost;
295      const IntVector  &_state;
296      const CostVector &_pi;
297      int &_in_arc;
298      int _search_arc_num;
299
300    public:
301
302      // Constructor
303      BestEligiblePivotRule(NetworkSimplex &ns) :
304        _source(ns._source), _target(ns._target),
305        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
306        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
307      {}
308
309      // Find next entering arc
310      bool findEnteringArc() {
311        Cost c, min = 0;
312        for (int e = 0; e < _search_arc_num; ++e) {
313          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
314          if (c < min) {
315            min = c;
316            _in_arc = e;
317          }
318        }
319        return min < 0;
320      }
321
322    }; //class BestEligiblePivotRule
323
324
325    // Implementation of the Block Search pivot rule
326    class BlockSearchPivotRule
327    {
328    private:
329
330      // References to the NetworkSimplex class
331      const IntVector  &_source;
332      const IntVector  &_target;
333      const CostVector &_cost;
334      const IntVector  &_state;
335      const CostVector &_pi;
336      int &_in_arc;
337      int _search_arc_num;
338
339      // Pivot rule data
340      int _block_size;
341      int _next_arc;
342
343    public:
344
345      // Constructor
346      BlockSearchPivotRule(NetworkSimplex &ns) :
347        _source(ns._source), _target(ns._target),
348        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
349        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
350        _next_arc(0)
351      {
352        // The main parameters of the pivot rule
353        const double BLOCK_SIZE_FACTOR = 0.5;
354        const int MIN_BLOCK_SIZE = 10;
355
356        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
357                                    std::sqrt(double(_search_arc_num))),
358                                MIN_BLOCK_SIZE );
359      }
360
361      // Find next entering arc
362      bool findEnteringArc() {
363        Cost c, min = 0;
364        int cnt = _block_size;
365        int e;
366        for (e = _next_arc; e < _search_arc_num; ++e) {
367          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
368          if (c < min) {
369            min = c;
370            _in_arc = e;
371          }
372          if (--cnt == 0) {
373            if (min < 0) goto search_end;
374            cnt = _block_size;
375          }
376        }
377        for (e = 0; e < _next_arc; ++e) {
378          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
379          if (c < min) {
380            min = c;
381            _in_arc = e;
382          }
383          if (--cnt == 0) {
384            if (min < 0) goto search_end;
385            cnt = _block_size;
386          }
387        }
388        if (min >= 0) return false;
389
390      search_end:
391        _next_arc = e;
392        return true;
393      }
394
395    }; //class BlockSearchPivotRule
396
397
398    // Implementation of the Candidate List pivot rule
399    class CandidateListPivotRule
400    {
401    private:
402
403      // References to the NetworkSimplex class
404      const IntVector  &_source;
405      const IntVector  &_target;
406      const CostVector &_cost;
407      const IntVector  &_state;
408      const CostVector &_pi;
409      int &_in_arc;
410      int _search_arc_num;
411
412      // Pivot rule data
413      IntVector _candidates;
414      int _list_length, _minor_limit;
415      int _curr_length, _minor_count;
416      int _next_arc;
417
418    public:
419
420      /// Constructor
421      CandidateListPivotRule(NetworkSimplex &ns) :
422        _source(ns._source), _target(ns._target),
423        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
424        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
425        _next_arc(0)
426      {
427        // The main parameters of the pivot rule
428        const double LIST_LENGTH_FACTOR = 0.25;
429        const int MIN_LIST_LENGTH = 10;
430        const double MINOR_LIMIT_FACTOR = 0.1;
431        const int MIN_MINOR_LIMIT = 3;
432
433        _list_length = std::max( int(LIST_LENGTH_FACTOR *
434                                     std::sqrt(double(_search_arc_num))),
435                                 MIN_LIST_LENGTH );
436        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
437                                 MIN_MINOR_LIMIT );
438        _curr_length = _minor_count = 0;
439        _candidates.resize(_list_length);
440      }
441
442      /// Find next entering arc
443      bool findEnteringArc() {
444        Cost min, c;
445        int e;
446        if (_curr_length > 0 && _minor_count < _minor_limit) {
447          // Minor iteration: select the best eligible arc from the
448          // current candidate list
449          ++_minor_count;
450          min = 0;
451          for (int i = 0; i < _curr_length; ++i) {
452            e = _candidates[i];
453            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
454            if (c < min) {
455              min = c;
456              _in_arc = e;
457            }
458            else if (c >= 0) {
459              _candidates[i--] = _candidates[--_curr_length];
460            }
461          }
462          if (min < 0) return true;
463        }
464
465        // Major iteration: build a new candidate list
466        min = 0;
467        _curr_length = 0;
468        for (e = _next_arc; e < _search_arc_num; ++e) {
469          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
470          if (c < 0) {
471            _candidates[_curr_length++] = e;
472            if (c < min) {
473              min = c;
474              _in_arc = e;
475            }
476            if (_curr_length == _list_length) goto search_end;
477          }
478        }
479        for (e = 0; e < _next_arc; ++e) {
480          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
481          if (c < 0) {
482            _candidates[_curr_length++] = e;
483            if (c < min) {
484              min = c;
485              _in_arc = e;
486            }
487            if (_curr_length == _list_length) goto search_end;
488          }
489        }
490        if (_curr_length == 0) return false;
491     
492      search_end:       
493        _minor_count = 1;
494        _next_arc = e;
495        return true;
496      }
497
498    }; //class CandidateListPivotRule
499
500
501    // Implementation of the Altering Candidate List pivot rule
502    class AlteringListPivotRule
503    {
504    private:
505
506      // References to the NetworkSimplex class
507      const IntVector  &_source;
508      const IntVector  &_target;
509      const CostVector &_cost;
510      const IntVector  &_state;
511      const CostVector &_pi;
512      int &_in_arc;
513      int _search_arc_num;
514
515      // Pivot rule data
516      int _block_size, _head_length, _curr_length;
517      int _next_arc;
518      IntVector _candidates;
519      CostVector _cand_cost;
520
521      // Functor class to compare arcs during sort of the candidate list
522      class SortFunc
523      {
524      private:
525        const CostVector &_map;
526      public:
527        SortFunc(const CostVector &map) : _map(map) {}
528        bool operator()(int left, int right) {
529          return _map[left] > _map[right];
530        }
531      };
532
533      SortFunc _sort_func;
534
535    public:
536
537      // Constructor
538      AlteringListPivotRule(NetworkSimplex &ns) :
539        _source(ns._source), _target(ns._target),
540        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
541        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
542        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
543      {
544        // The main parameters of the pivot rule
545        const double BLOCK_SIZE_FACTOR = 1.0;
546        const int MIN_BLOCK_SIZE = 10;
547        const double HEAD_LENGTH_FACTOR = 0.1;
548        const int MIN_HEAD_LENGTH = 3;
549
550        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
551                                    std::sqrt(double(_search_arc_num))),
552                                MIN_BLOCK_SIZE );
553        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
554                                 MIN_HEAD_LENGTH );
555        _candidates.resize(_head_length + _block_size);
556        _curr_length = 0;
557      }
558
559      // Find next entering arc
560      bool findEnteringArc() {
561        // Check the current candidate list
562        int e;
563        for (int i = 0; i < _curr_length; ++i) {
564          e = _candidates[i];
565          _cand_cost[e] = _state[e] *
566            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
567          if (_cand_cost[e] >= 0) {
568            _candidates[i--] = _candidates[--_curr_length];
569          }
570        }
571
572        // Extend the list
573        int cnt = _block_size;
574        int limit = _head_length;
575
576        for (e = _next_arc; e < _search_arc_num; ++e) {
577          _cand_cost[e] = _state[e] *
578            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
579          if (_cand_cost[e] < 0) {
580            _candidates[_curr_length++] = e;
581          }
582          if (--cnt == 0) {
583            if (_curr_length > limit) goto search_end;
584            limit = 0;
585            cnt = _block_size;
586          }
587        }
588        for (e = 0; e < _next_arc; ++e) {
589          _cand_cost[e] = _state[e] *
590            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
591          if (_cand_cost[e] < 0) {
592            _candidates[_curr_length++] = e;
593          }
594          if (--cnt == 0) {
595            if (_curr_length > limit) goto search_end;
596            limit = 0;
597            cnt = _block_size;
598          }
599        }
600        if (_curr_length == 0) return false;
601       
602      search_end:
603
604        // Make heap of the candidate list (approximating a partial sort)
605        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
606                   _sort_func );
607
608        // Pop the first element of the heap
609        _in_arc = _candidates[0];
610        _next_arc = e;
611        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
612                  _sort_func );
613        _curr_length = std::min(_head_length, _curr_length - 1);
614        return true;
615      }
616
617    }; //class AlteringListPivotRule
618
619  public:
620
621    /// \brief Constructor.
622    ///
623    /// The constructor of the class.
624    ///
625    /// \param graph The digraph the algorithm runs on.
626    /// \param arc_mixing Indicate if the arcs have to be stored in a
627    /// mixed order in the internal data structure.
628    /// In special cases, it could lead to better overall performance,
629    /// but it is usually slower. Therefore it is disabled by default.
630    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
631      _graph(graph), _node_id(graph), _arc_id(graph),
632      INF(std::numeric_limits<Value>::has_infinity ?
633          std::numeric_limits<Value>::infinity() :
634          std::numeric_limits<Value>::max())
635    {
636      // Check the value types
637      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
638        "The flow type of NetworkSimplex must be signed");
639      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
640        "The cost type of NetworkSimplex must be signed");
641       
642      // Resize vectors
643      _node_num = countNodes(_graph);
644      _arc_num = countArcs(_graph);
645      int all_node_num = _node_num + 1;
646      int max_arc_num = _arc_num + 2 * _node_num;
647
648      _source.resize(max_arc_num);
649      _target.resize(max_arc_num);
650
651      _lower.resize(_arc_num);
652      _upper.resize(_arc_num);
653      _cap.resize(max_arc_num);
654      _cost.resize(max_arc_num);
655      _supply.resize(all_node_num);
656      _flow.resize(max_arc_num);
657      _pi.resize(all_node_num);
658
659      _parent.resize(all_node_num);
660      _pred.resize(all_node_num);
661      _forward.resize(all_node_num);
662      _thread.resize(all_node_num);
663      _rev_thread.resize(all_node_num);
664      _succ_num.resize(all_node_num);
665      _last_succ.resize(all_node_num);
666      _state.resize(max_arc_num);
667
668      // Copy the graph
669      int i = 0;
670      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
671        _node_id[n] = i;
672      }
673      if (arc_mixing) {
674        // Store the arcs in a mixed order
675        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
676        int i = 0, j = 0;
677        for (ArcIt a(_graph); a != INVALID; ++a) {
678          _arc_id[a] = i;
679          _source[i] = _node_id[_graph.source(a)];
680          _target[i] = _node_id[_graph.target(a)];
681          if ((i += k) >= _arc_num) i = ++j;
682        }
683      } else {
684        // Store the arcs in the original order
685        int i = 0;
686        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
687          _arc_id[a] = i;
688          _source[i] = _node_id[_graph.source(a)];
689          _target[i] = _node_id[_graph.target(a)];
690        }
691      }
692     
693      // Reset parameters
694      reset();
695    }
696
697    /// \name Parameters
698    /// The parameters of the algorithm can be specified using these
699    /// functions.
700
701    /// @{
702
703    /// \brief Set the lower bounds on the arcs.
704    ///
705    /// This function sets the lower bounds on the arcs.
706    /// If it is not used before calling \ref run(), the lower bounds
707    /// will be set to zero on all arcs.
708    ///
709    /// \param map An arc map storing the lower bounds.
710    /// Its \c Value type must be convertible to the \c Value type
711    /// of the algorithm.
712    ///
713    /// \return <tt>(*this)</tt>
714    template <typename LowerMap>
715    NetworkSimplex& lowerMap(const LowerMap& map) {
716      _have_lower = true;
717      for (ArcIt a(_graph); a != INVALID; ++a) {
718        _lower[_arc_id[a]] = map[a];
719      }
720      return *this;
721    }
722
723    /// \brief Set the upper bounds (capacities) on the arcs.
724    ///
725    /// This function sets the upper bounds (capacities) on the arcs.
726    /// If it is not used before calling \ref run(), the upper bounds
727    /// will be set to \ref INF on all arcs (i.e. the flow value will be
728    /// unbounded from above on each arc).
729    ///
730    /// \param map An arc map storing the upper bounds.
731    /// Its \c Value type must be convertible to the \c Value type
732    /// of the algorithm.
733    ///
734    /// \return <tt>(*this)</tt>
735    template<typename UpperMap>
736    NetworkSimplex& upperMap(const UpperMap& map) {
737      for (ArcIt a(_graph); a != INVALID; ++a) {
738        _upper[_arc_id[a]] = map[a];
739      }
740      return *this;
741    }
742
743    /// \brief Set the costs of the arcs.
744    ///
745    /// This function sets the costs of the arcs.
746    /// If it is not used before calling \ref run(), the costs
747    /// will be set to \c 1 on all arcs.
748    ///
749    /// \param map An arc map storing the costs.
750    /// Its \c Value type must be convertible to the \c Cost type
751    /// of the algorithm.
752    ///
753    /// \return <tt>(*this)</tt>
754    template<typename CostMap>
755    NetworkSimplex& costMap(const CostMap& map) {
756      for (ArcIt a(_graph); a != INVALID; ++a) {
757        _cost[_arc_id[a]] = map[a];
758      }
759      return *this;
760    }
761
762    /// \brief Set the supply values of the nodes.
763    ///
764    /// This function sets the supply values of the nodes.
765    /// If neither this function nor \ref stSupply() is used before
766    /// calling \ref run(), the supply of each node will be set to zero.
767    ///
768    /// \param map A node map storing the supply values.
769    /// Its \c Value type must be convertible to the \c Value type
770    /// of the algorithm.
771    ///
772    /// \return <tt>(*this)</tt>
773    template<typename SupplyMap>
774    NetworkSimplex& supplyMap(const SupplyMap& map) {
775      for (NodeIt n(_graph); n != INVALID; ++n) {
776        _supply[_node_id[n]] = map[n];
777      }
778      return *this;
779    }
780
781    /// \brief Set single source and target nodes and a supply value.
782    ///
783    /// This function sets a single source node and a single target node
784    /// and the required flow value.
785    /// If neither this function nor \ref supplyMap() is used before
786    /// calling \ref run(), the supply of each node will be set to zero.
787    ///
788    /// Using this function has the same effect as using \ref supplyMap()
789    /// with such a map in which \c k is assigned to \c s, \c -k is
790    /// assigned to \c t and all other nodes have zero supply value.
791    ///
792    /// \param s The source node.
793    /// \param t The target node.
794    /// \param k The required amount of flow from node \c s to node \c t
795    /// (i.e. the supply of \c s and the demand of \c t).
796    ///
797    /// \return <tt>(*this)</tt>
798    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
799      for (int i = 0; i != _node_num; ++i) {
800        _supply[i] = 0;
801      }
802      _supply[_node_id[s]] =  k;
803      _supply[_node_id[t]] = -k;
804      return *this;
805    }
806   
807    /// \brief Set the type of the supply constraints.
808    ///
809    /// This function sets the type of the supply/demand constraints.
810    /// If it is not used before calling \ref run(), the \ref GEQ supply
811    /// type will be used.
812    ///
813    /// For more information see \ref SupplyType.
814    ///
815    /// \return <tt>(*this)</tt>
816    NetworkSimplex& supplyType(SupplyType supply_type) {
817      _stype = supply_type;
818      return *this;
819    }
820
821    /// @}
822
823    /// \name Execution Control
824    /// The algorithm can be executed using \ref run().
825
826    /// @{
827
828    /// \brief Run the algorithm.
829    ///
830    /// This function runs the algorithm.
831    /// The paramters can be specified using functions \ref lowerMap(),
832    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
833    /// \ref supplyType().
834    /// For example,
835    /// \code
836    ///   NetworkSimplex<ListDigraph> ns(graph);
837    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
838    ///     .supplyMap(sup).run();
839    /// \endcode
840    ///
841    /// This function can be called more than once. All the parameters
842    /// that have been given are kept for the next call, unless
843    /// \ref reset() is called, thus only the modified parameters
844    /// have to be set again. See \ref reset() for examples.
845    /// However the underlying digraph must not be modified after this
846    /// class have been constructed, since it copies and extends the graph.
847    ///
848    /// \param pivot_rule The pivot rule that will be used during the
849    /// algorithm. For more information see \ref PivotRule.
850    ///
851    /// \return \c INFEASIBLE if no feasible flow exists,
852    /// \n \c OPTIMAL if the problem has optimal solution
853    /// (i.e. it is feasible and bounded), and the algorithm has found
854    /// optimal flow and node potentials (primal and dual solutions),
855    /// \n \c UNBOUNDED if the objective function of the problem is
856    /// unbounded, i.e. there is a directed cycle having negative total
857    /// cost and infinite upper bound.
858    ///
859    /// \see ProblemType, PivotRule
860    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
861      if (!init()) return INFEASIBLE;
862      return start(pivot_rule);
863    }
864
865    /// \brief Reset all the parameters that have been given before.
866    ///
867    /// This function resets all the paramaters that have been given
868    /// before using functions \ref lowerMap(), \ref upperMap(),
869    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
870    ///
871    /// It is useful for multiple run() calls. If this function is not
872    /// used, all the parameters given before are kept for the next
873    /// \ref run() call.
874    /// However the underlying digraph must not be modified after this
875    /// class have been constructed, since it copies and extends the graph.
876    ///
877    /// For example,
878    /// \code
879    ///   NetworkSimplex<ListDigraph> ns(graph);
880    ///
881    ///   // First run
882    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
883    ///     .supplyMap(sup).run();
884    ///
885    ///   // Run again with modified cost map (reset() is not called,
886    ///   // so only the cost map have to be set again)
887    ///   cost[e] += 100;
888    ///   ns.costMap(cost).run();
889    ///
890    ///   // Run again from scratch using reset()
891    ///   // (the lower bounds will be set to zero on all arcs)
892    ///   ns.reset();
893    ///   ns.upperMap(capacity).costMap(cost)
894    ///     .supplyMap(sup).run();
895    /// \endcode
896    ///
897    /// \return <tt>(*this)</tt>
898    NetworkSimplex& reset() {
899      for (int i = 0; i != _node_num; ++i) {
900        _supply[i] = 0;
901      }
902      for (int i = 0; i != _arc_num; ++i) {
903        _lower[i] = 0;
904        _upper[i] = INF;
905        _cost[i] = 1;
906      }
907      _have_lower = false;
908      _stype = GEQ;
909      return *this;
910    }
911
912    /// @}
913
914    /// \name Query Functions
915    /// The results of the algorithm can be obtained using these
916    /// functions.\n
917    /// The \ref run() function must be called before using them.
918
919    /// @{
920
921    /// \brief Return the total cost of the found flow.
922    ///
923    /// This function returns the total cost of the found flow.
924    /// Its complexity is O(e).
925    ///
926    /// \note The return type of the function can be specified as a
927    /// template parameter. For example,
928    /// \code
929    ///   ns.totalCost<double>();
930    /// \endcode
931    /// It is useful if the total cost cannot be stored in the \c Cost
932    /// type of the algorithm, which is the default return type of the
933    /// function.
934    ///
935    /// \pre \ref run() must be called before using this function.
936    template <typename Number>
937    Number totalCost() const {
938      Number c = 0;
939      for (ArcIt a(_graph); a != INVALID; ++a) {
940        int i = _arc_id[a];
941        c += Number(_flow[i]) * Number(_cost[i]);
942      }
943      return c;
944    }
945
946#ifndef DOXYGEN
947    Cost totalCost() const {
948      return totalCost<Cost>();
949    }
950#endif
951
952    /// \brief Return the flow on the given arc.
953    ///
954    /// This function returns the flow on the given arc.
955    ///
956    /// \pre \ref run() must be called before using this function.
957    Value flow(const Arc& a) const {
958      return _flow[_arc_id[a]];
959    }
960
961    /// \brief Return the flow map (the primal solution).
962    ///
963    /// This function copies the flow value on each arc into the given
964    /// map. The \c Value type of the algorithm must be convertible to
965    /// the \c Value type of the map.
966    ///
967    /// \pre \ref run() must be called before using this function.
968    template <typename FlowMap>
969    void flowMap(FlowMap &map) const {
970      for (ArcIt a(_graph); a != INVALID; ++a) {
971        map.set(a, _flow[_arc_id[a]]);
972      }
973    }
974
975    /// \brief Return the potential (dual value) of the given node.
976    ///
977    /// This function returns the potential (dual value) of the
978    /// given node.
979    ///
980    /// \pre \ref run() must be called before using this function.
981    Cost potential(const Node& n) const {
982      return _pi[_node_id[n]];
983    }
984
985    /// \brief Return the potential map (the dual solution).
986    ///
987    /// This function copies the potential (dual value) of each node
988    /// into the given map.
989    /// The \c Cost type of the algorithm must be convertible to the
990    /// \c Value type of the map.
991    ///
992    /// \pre \ref run() must be called before using this function.
993    template <typename PotentialMap>
994    void potentialMap(PotentialMap &map) const {
995      for (NodeIt n(_graph); n != INVALID; ++n) {
996        map.set(n, _pi[_node_id[n]]);
997      }
998    }
999
1000    /// @}
1001
1002  private:
1003
1004    // Initialize internal data structures
1005    bool init() {
1006      if (_node_num == 0) return false;
1007
1008      // Check the sum of supply values
1009      _sum_supply = 0;
1010      for (int i = 0; i != _node_num; ++i) {
1011        _sum_supply += _supply[i];
1012      }
1013      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1014             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1015
1016      // Remove non-zero lower bounds
1017      if (_have_lower) {
1018        for (int i = 0; i != _arc_num; ++i) {
1019          Value c = _lower[i];
1020          if (c >= 0) {
1021            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1022          } else {
1023            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1024          }
1025          _supply[_source[i]] -= c;
1026          _supply[_target[i]] += c;
1027        }
1028      } else {
1029        for (int i = 0; i != _arc_num; ++i) {
1030          _cap[i] = _upper[i];
1031        }
1032      }
1033
1034      // Initialize artifical cost
1035      Cost ART_COST;
1036      if (std::numeric_limits<Cost>::is_exact) {
1037        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1038      } else {
1039        ART_COST = std::numeric_limits<Cost>::min();
1040        for (int i = 0; i != _arc_num; ++i) {
1041          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1042        }
1043        ART_COST = (ART_COST + 1) * _node_num;
1044      }
1045
1046      // Initialize arc maps
1047      for (int i = 0; i != _arc_num; ++i) {
1048        _flow[i] = 0;
1049        _state[i] = STATE_LOWER;
1050      }
1051     
1052      // Set data for the artificial root node
1053      _root = _node_num;
1054      _parent[_root] = -1;
1055      _pred[_root] = -1;
1056      _thread[_root] = 0;
1057      _rev_thread[0] = _root;
1058      _succ_num[_root] = _node_num + 1;
1059      _last_succ[_root] = _root - 1;
1060      _supply[_root] = -_sum_supply;
1061      _pi[_root] = 0;
1062
1063      // Add artificial arcs and initialize the spanning tree data structure
1064      if (_sum_supply == 0) {
1065        // EQ supply constraints
1066        _search_arc_num = _arc_num;
1067        _all_arc_num = _arc_num + _node_num;
1068        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1069          _parent[u] = _root;
1070          _pred[u] = e;
1071          _thread[u] = u + 1;
1072          _rev_thread[u + 1] = u;
1073          _succ_num[u] = 1;
1074          _last_succ[u] = u;
1075          _cap[e] = INF;
1076          _state[e] = STATE_TREE;
1077          if (_supply[u] >= 0) {
1078            _forward[u] = true;
1079            _pi[u] = 0;
1080            _source[e] = u;
1081            _target[e] = _root;
1082            _flow[e] = _supply[u];
1083            _cost[e] = 0;
1084          } else {
1085            _forward[u] = false;
1086            _pi[u] = ART_COST;
1087            _source[e] = _root;
1088            _target[e] = u;
1089            _flow[e] = -_supply[u];
1090            _cost[e] = ART_COST;
1091          }
1092        }
1093      }
1094      else if (_sum_supply > 0) {
1095        // LEQ supply constraints
1096        _search_arc_num = _arc_num + _node_num;
1097        int f = _arc_num + _node_num;
1098        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1099          _parent[u] = _root;
1100          _thread[u] = u + 1;
1101          _rev_thread[u + 1] = u;
1102          _succ_num[u] = 1;
1103          _last_succ[u] = u;
1104          if (_supply[u] >= 0) {
1105            _forward[u] = true;
1106            _pi[u] = 0;
1107            _pred[u] = e;
1108            _source[e] = u;
1109            _target[e] = _root;
1110            _cap[e] = INF;
1111            _flow[e] = _supply[u];
1112            _cost[e] = 0;
1113            _state[e] = STATE_TREE;
1114          } else {
1115            _forward[u] = false;
1116            _pi[u] = ART_COST;
1117            _pred[u] = f;
1118            _source[f] = _root;
1119            _target[f] = u;
1120            _cap[f] = INF;
1121            _flow[f] = -_supply[u];
1122            _cost[f] = ART_COST;
1123            _state[f] = STATE_TREE;
1124            _source[e] = u;
1125            _target[e] = _root;
1126            _cap[e] = INF;
1127            _flow[e] = 0;
1128            _cost[e] = 0;
1129            _state[e] = STATE_LOWER;
1130            ++f;
1131          }
1132        }
1133        _all_arc_num = f;
1134      }
1135      else {
1136        // GEQ supply constraints
1137        _search_arc_num = _arc_num + _node_num;
1138        int f = _arc_num + _node_num;
1139        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1140          _parent[u] = _root;
1141          _thread[u] = u + 1;
1142          _rev_thread[u + 1] = u;
1143          _succ_num[u] = 1;
1144          _last_succ[u] = u;
1145          if (_supply[u] <= 0) {
1146            _forward[u] = false;
1147            _pi[u] = 0;
1148            _pred[u] = e;
1149            _source[e] = _root;
1150            _target[e] = u;
1151            _cap[e] = INF;
1152            _flow[e] = -_supply[u];
1153            _cost[e] = 0;
1154            _state[e] = STATE_TREE;
1155          } else {
1156            _forward[u] = true;
1157            _pi[u] = -ART_COST;
1158            _pred[u] = f;
1159            _source[f] = u;
1160            _target[f] = _root;
1161            _cap[f] = INF;
1162            _flow[f] = _supply[u];
1163            _state[f] = STATE_TREE;
1164            _cost[f] = ART_COST;
1165            _source[e] = _root;
1166            _target[e] = u;
1167            _cap[e] = INF;
1168            _flow[e] = 0;
1169            _cost[e] = 0;
1170            _state[e] = STATE_LOWER;
1171            ++f;
1172          }
1173        }
1174        _all_arc_num = f;
1175      }
1176
1177      return true;
1178    }
1179
1180    // Find the join node
1181    void findJoinNode() {
1182      int u = _source[in_arc];
1183      int v = _target[in_arc];
1184      while (u != v) {
1185        if (_succ_num[u] < _succ_num[v]) {
1186          u = _parent[u];
1187        } else {
1188          v = _parent[v];
1189        }
1190      }
1191      join = u;
1192    }
1193
1194    // Find the leaving arc of the cycle and returns true if the
1195    // leaving arc is not the same as the entering arc
1196    bool findLeavingArc() {
1197      // Initialize first and second nodes according to the direction
1198      // of the cycle
1199      if (_state[in_arc] == STATE_LOWER) {
1200        first  = _source[in_arc];
1201        second = _target[in_arc];
1202      } else {
1203        first  = _target[in_arc];
1204        second = _source[in_arc];
1205      }
1206      delta = _cap[in_arc];
1207      int result = 0;
1208      Value d;
1209      int e;
1210
1211      // Search the cycle along the path form the first node to the root
1212      for (int u = first; u != join; u = _parent[u]) {
1213        e = _pred[u];
1214        d = _forward[u] ?
1215          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1216        if (d < delta) {
1217          delta = d;
1218          u_out = u;
1219          result = 1;
1220        }
1221      }
1222      // Search the cycle along the path form the second node to the root
1223      for (int u = second; u != join; u = _parent[u]) {
1224        e = _pred[u];
1225        d = _forward[u] ?
1226          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1227        if (d <= delta) {
1228          delta = d;
1229          u_out = u;
1230          result = 2;
1231        }
1232      }
1233
1234      if (result == 1) {
1235        u_in = first;
1236        v_in = second;
1237      } else {
1238        u_in = second;
1239        v_in = first;
1240      }
1241      return result != 0;
1242    }
1243
1244    // Change _flow and _state vectors
1245    void changeFlow(bool change) {
1246      // Augment along the cycle
1247      if (delta > 0) {
1248        Value val = _state[in_arc] * delta;
1249        _flow[in_arc] += val;
1250        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1251          _flow[_pred[u]] += _forward[u] ? -val : val;
1252        }
1253        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1254          _flow[_pred[u]] += _forward[u] ? val : -val;
1255        }
1256      }
1257      // Update the state of the entering and leaving arcs
1258      if (change) {
1259        _state[in_arc] = STATE_TREE;
1260        _state[_pred[u_out]] =
1261          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1262      } else {
1263        _state[in_arc] = -_state[in_arc];
1264      }
1265    }
1266
1267    // Update the tree structure
1268    void updateTreeStructure() {
1269      int u, w;
1270      int old_rev_thread = _rev_thread[u_out];
1271      int old_succ_num = _succ_num[u_out];
1272      int old_last_succ = _last_succ[u_out];
1273      v_out = _parent[u_out];
1274
1275      u = _last_succ[u_in];  // the last successor of u_in
1276      right = _thread[u];    // the node after it
1277
1278      // Handle the case when old_rev_thread equals to v_in
1279      // (it also means that join and v_out coincide)
1280      if (old_rev_thread == v_in) {
1281        last = _thread[_last_succ[u_out]];
1282      } else {
1283        last = _thread[v_in];
1284      }
1285
1286      // Update _thread and _parent along the stem nodes (i.e. the nodes
1287      // between u_in and u_out, whose parent have to be changed)
1288      _thread[v_in] = stem = u_in;
1289      _dirty_revs.clear();
1290      _dirty_revs.push_back(v_in);
1291      par_stem = v_in;
1292      while (stem != u_out) {
1293        // Insert the next stem node into the thread list
1294        new_stem = _parent[stem];
1295        _thread[u] = new_stem;
1296        _dirty_revs.push_back(u);
1297
1298        // Remove the subtree of stem from the thread list
1299        w = _rev_thread[stem];
1300        _thread[w] = right;
1301        _rev_thread[right] = w;
1302
1303        // Change the parent node and shift stem nodes
1304        _parent[stem] = par_stem;
1305        par_stem = stem;
1306        stem = new_stem;
1307
1308        // Update u and right
1309        u = _last_succ[stem] == _last_succ[par_stem] ?
1310          _rev_thread[par_stem] : _last_succ[stem];
1311        right = _thread[u];
1312      }
1313      _parent[u_out] = par_stem;
1314      _thread[u] = last;
1315      _rev_thread[last] = u;
1316      _last_succ[u_out] = u;
1317
1318      // Remove the subtree of u_out from the thread list except for
1319      // the case when old_rev_thread equals to v_in
1320      // (it also means that join and v_out coincide)
1321      if (old_rev_thread != v_in) {
1322        _thread[old_rev_thread] = right;
1323        _rev_thread[right] = old_rev_thread;
1324      }
1325
1326      // Update _rev_thread using the new _thread values
1327      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1328        u = _dirty_revs[i];
1329        _rev_thread[_thread[u]] = u;
1330      }
1331
1332      // Update _pred, _forward, _last_succ and _succ_num for the
1333      // stem nodes from u_out to u_in
1334      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1335      u = u_out;
1336      while (u != u_in) {
1337        w = _parent[u];
1338        _pred[u] = _pred[w];
1339        _forward[u] = !_forward[w];
1340        tmp_sc += _succ_num[u] - _succ_num[w];
1341        _succ_num[u] = tmp_sc;
1342        _last_succ[w] = tmp_ls;
1343        u = w;
1344      }
1345      _pred[u_in] = in_arc;
1346      _forward[u_in] = (u_in == _source[in_arc]);
1347      _succ_num[u_in] = old_succ_num;
1348
1349      // Set limits for updating _last_succ form v_in and v_out
1350      // towards the root
1351      int up_limit_in = -1;
1352      int up_limit_out = -1;
1353      if (_last_succ[join] == v_in) {
1354        up_limit_out = join;
1355      } else {
1356        up_limit_in = join;
1357      }
1358
1359      // Update _last_succ from v_in towards the root
1360      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1361           u = _parent[u]) {
1362        _last_succ[u] = _last_succ[u_out];
1363      }
1364      // Update _last_succ from v_out towards the root
1365      if (join != old_rev_thread && v_in != old_rev_thread) {
1366        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1367             u = _parent[u]) {
1368          _last_succ[u] = old_rev_thread;
1369        }
1370      } else {
1371        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1372             u = _parent[u]) {
1373          _last_succ[u] = _last_succ[u_out];
1374        }
1375      }
1376
1377      // Update _succ_num from v_in to join
1378      for (u = v_in; u != join; u = _parent[u]) {
1379        _succ_num[u] += old_succ_num;
1380      }
1381      // Update _succ_num from v_out to join
1382      for (u = v_out; u != join; u = _parent[u]) {
1383        _succ_num[u] -= old_succ_num;
1384      }
1385    }
1386
1387    // Update potentials
1388    void updatePotential() {
1389      Cost sigma = _forward[u_in] ?
1390        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1391        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1392      // Update potentials in the subtree, which has been moved
1393      int end = _thread[_last_succ[u_in]];
1394      for (int u = u_in; u != end; u = _thread[u]) {
1395        _pi[u] += sigma;
1396      }
1397    }
1398
1399    // Execute the algorithm
1400    ProblemType start(PivotRule pivot_rule) {
1401      // Select the pivot rule implementation
1402      switch (pivot_rule) {
1403        case FIRST_ELIGIBLE:
1404          return start<FirstEligiblePivotRule>();
1405        case BEST_ELIGIBLE:
1406          return start<BestEligiblePivotRule>();
1407        case BLOCK_SEARCH:
1408          return start<BlockSearchPivotRule>();
1409        case CANDIDATE_LIST:
1410          return start<CandidateListPivotRule>();
1411        case ALTERING_LIST:
1412          return start<AlteringListPivotRule>();
1413      }
1414      return INFEASIBLE; // avoid warning
1415    }
1416
1417    template <typename PivotRuleImpl>
1418    ProblemType start() {
1419      PivotRuleImpl pivot(*this);
1420
1421      // Execute the Network Simplex algorithm
1422      while (pivot.findEnteringArc()) {
1423        findJoinNode();
1424        bool change = findLeavingArc();
1425        if (delta >= INF) return UNBOUNDED;
1426        changeFlow(change);
1427        if (change) {
1428          updateTreeStructure();
1429          updatePotential();
1430        }
1431      }
1432     
1433      // Check feasibility
1434      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1435        if (_flow[e] != 0) return INFEASIBLE;
1436      }
1437
1438      // Transform the solution and the supply map to the original form
1439      if (_have_lower) {
1440        for (int i = 0; i != _arc_num; ++i) {
1441          Value c = _lower[i];
1442          if (c != 0) {
1443            _flow[i] += c;
1444            _supply[_source[i]] += c;
1445            _supply[_target[i]] -= c;
1446          }
1447        }
1448      }
1449     
1450      // Shift potentials to meet the requirements of the GEQ/LEQ type
1451      // optimality conditions
1452      if (_sum_supply == 0) {
1453        if (_stype == GEQ) {
1454          Cost max_pot = std::numeric_limits<Cost>::min();
1455          for (int i = 0; i != _node_num; ++i) {
1456            if (_pi[i] > max_pot) max_pot = _pi[i];
1457          }
1458          if (max_pot > 0) {
1459            for (int i = 0; i != _node_num; ++i)
1460              _pi[i] -= max_pot;
1461          }
1462        } else {
1463          Cost min_pot = std::numeric_limits<Cost>::max();
1464          for (int i = 0; i != _node_num; ++i) {
1465            if (_pi[i] < min_pot) min_pot = _pi[i];
1466          }
1467          if (min_pot < 0) {
1468            for (int i = 0; i != _node_num; ++i)
1469              _pi[i] -= min_pot;
1470          }
1471        }
1472      }
1473
1474      return OPTIMAL;
1475    }
1476
1477  }; //class NetworkSimplex
1478
1479  ///@}
1480
1481} //namespace lemon
1482
1483#endif //LEMON_NETWORK_SIMPLEX_H
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