COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 774:cab85bd7859b

Last change on this file since 774:cab85bd7859b was 774:cab85bd7859b, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Small improvements in NS pivot rules (#298)

File size: 45.1 KB
Line 
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<Arc> ArcVector;
165    typedef std::vector<Node> NodeVector;
166    typedef std::vector<int> IntVector;
167    typedef std::vector<bool> BoolVector;
168    typedef std::vector<Value> ValueVector;
169    typedef std::vector<Cost> CostVector;
170
171    // State constants for arcs
172    enum ArcStateEnum {
173      STATE_UPPER = -1,
174      STATE_TREE  =  0,
175      STATE_LOWER =  1
176    };
177
178  private:
179
180    // Data related to the underlying digraph
181    const GR &_graph;
182    int _node_num;
183    int _arc_num;
184    int _all_arc_num;
185    int _search_arc_num;
186
187    // Parameters of the problem
188    bool _have_lower;
189    SupplyType _stype;
190    Value _sum_supply;
191
192    // Data structures for storing the digraph
193    IntNodeMap _node_id;
194    IntArcMap _arc_id;
195    IntVector _source;
196    IntVector _target;
197
198    // Node and arc data
199    ValueVector _lower;
200    ValueVector _upper;
201    ValueVector _cap;
202    CostVector _cost;
203    ValueVector _supply;
204    ValueVector _flow;
205    CostVector _pi;
206
207    // Data for storing the spanning tree structure
208    IntVector _parent;
209    IntVector _pred;
210    IntVector _thread;
211    IntVector _rev_thread;
212    IntVector _succ_num;
213    IntVector _last_succ;
214    IntVector _dirty_revs;
215    BoolVector _forward;
216    IntVector _state;
217    int _root;
218
219    // Temporary data used in the current pivot iteration
220    int in_arc, join, u_in, v_in, u_out, v_out;
221    int first, second, right, last;
222    int stem, par_stem, new_stem;
223    Value delta;
224
225  public:
226 
227    /// \brief Constant for infinite upper bounds (capacities).
228    ///
229    /// Constant for infinite upper bounds (capacities).
230    /// It is \c std::numeric_limits<Value>::infinity() if available,
231    /// \c std::numeric_limits<Value>::max() otherwise.
232    const Value INF;
233
234  private:
235
236    // Implementation of the First Eligible pivot rule
237    class FirstEligiblePivotRule
238    {
239    private:
240
241      // References to the NetworkSimplex class
242      const IntVector  &_source;
243      const IntVector  &_target;
244      const CostVector &_cost;
245      const IntVector  &_state;
246      const CostVector &_pi;
247      int &_in_arc;
248      int _search_arc_num;
249
250      // Pivot rule data
251      int _next_arc;
252
253    public:
254
255      // Constructor
256      FirstEligiblePivotRule(NetworkSimplex &ns) :
257        _source(ns._source), _target(ns._target),
258        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
259        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
260        _next_arc(0)
261      {}
262
263      // Find next entering arc
264      bool findEnteringArc() {
265        Cost c;
266        for (int e = _next_arc; e < _search_arc_num; ++e) {
267          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
268          if (c < 0) {
269            _in_arc = e;
270            _next_arc = e + 1;
271            return true;
272          }
273        }
274        for (int e = 0; e < _next_arc; ++e) {
275          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
276          if (c < 0) {
277            _in_arc = e;
278            _next_arc = e + 1;
279            return true;
280          }
281        }
282        return false;
283      }
284
285    }; //class FirstEligiblePivotRule
286
287
288    // Implementation of the Best Eligible pivot rule
289    class BestEligiblePivotRule
290    {
291    private:
292
293      // References to the NetworkSimplex class
294      const IntVector  &_source;
295      const IntVector  &_target;
296      const CostVector &_cost;
297      const IntVector  &_state;
298      const CostVector &_pi;
299      int &_in_arc;
300      int _search_arc_num;
301
302    public:
303
304      // Constructor
305      BestEligiblePivotRule(NetworkSimplex &ns) :
306        _source(ns._source), _target(ns._target),
307        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
308        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
309      {}
310
311      // Find next entering arc
312      bool findEnteringArc() {
313        Cost c, min = 0;
314        for (int e = 0; e < _search_arc_num; ++e) {
315          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
316          if (c < min) {
317            min = c;
318            _in_arc = e;
319          }
320        }
321        return min < 0;
322      }
323
324    }; //class BestEligiblePivotRule
325
326
327    // Implementation of the Block Search pivot rule
328    class BlockSearchPivotRule
329    {
330    private:
331
332      // References to the NetworkSimplex class
333      const IntVector  &_source;
334      const IntVector  &_target;
335      const CostVector &_cost;
336      const IntVector  &_state;
337      const CostVector &_pi;
338      int &_in_arc;
339      int _search_arc_num;
340
341      // Pivot rule data
342      int _block_size;
343      int _next_arc;
344
345    public:
346
347      // Constructor
348      BlockSearchPivotRule(NetworkSimplex &ns) :
349        _source(ns._source), _target(ns._target),
350        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
351        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
352        _next_arc(0)
353      {
354        // The main parameters of the pivot rule
355        const double BLOCK_SIZE_FACTOR = 0.5;
356        const int MIN_BLOCK_SIZE = 10;
357
358        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
359                                    std::sqrt(double(_search_arc_num))),
360                                MIN_BLOCK_SIZE );
361      }
362
363      // Find next entering arc
364      bool findEnteringArc() {
365        Cost c, min = 0;
366        int cnt = _block_size;
367        int e;
368        for (e = _next_arc; e < _search_arc_num; ++e) {
369          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
370          if (c < min) {
371            min = c;
372            _in_arc = e;
373          }
374          if (--cnt == 0) {
375            if (min < 0) goto search_end;
376            cnt = _block_size;
377          }
378        }
379        for (e = 0; e < _next_arc; ++e) {
380          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
381          if (c < min) {
382            min = c;
383            _in_arc = e;
384          }
385          if (--cnt == 0) {
386            if (min < 0) goto search_end;
387            cnt = _block_size;
388          }
389        }
390        if (min >= 0) return false;
391
392      search_end:
393        _next_arc = e;
394        return true;
395      }
396
397    }; //class BlockSearchPivotRule
398
399
400    // Implementation of the Candidate List pivot rule
401    class CandidateListPivotRule
402    {
403    private:
404
405      // References to the NetworkSimplex class
406      const IntVector  &_source;
407      const IntVector  &_target;
408      const CostVector &_cost;
409      const IntVector  &_state;
410      const CostVector &_pi;
411      int &_in_arc;
412      int _search_arc_num;
413
414      // Pivot rule data
415      IntVector _candidates;
416      int _list_length, _minor_limit;
417      int _curr_length, _minor_count;
418      int _next_arc;
419
420    public:
421
422      /// Constructor
423      CandidateListPivotRule(NetworkSimplex &ns) :
424        _source(ns._source), _target(ns._target),
425        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
426        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
427        _next_arc(0)
428      {
429        // The main parameters of the pivot rule
430        const double LIST_LENGTH_FACTOR = 0.25;
431        const int MIN_LIST_LENGTH = 10;
432        const double MINOR_LIMIT_FACTOR = 0.1;
433        const int MIN_MINOR_LIMIT = 3;
434
435        _list_length = std::max( int(LIST_LENGTH_FACTOR *
436                                     std::sqrt(double(_search_arc_num))),
437                                 MIN_LIST_LENGTH );
438        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
439                                 MIN_MINOR_LIMIT );
440        _curr_length = _minor_count = 0;
441        _candidates.resize(_list_length);
442      }
443
444      /// Find next entering arc
445      bool findEnteringArc() {
446        Cost min, c;
447        int e;
448        if (_curr_length > 0 && _minor_count < _minor_limit) {
449          // Minor iteration: select the best eligible arc from the
450          // current candidate list
451          ++_minor_count;
452          min = 0;
453          for (int i = 0; i < _curr_length; ++i) {
454            e = _candidates[i];
455            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
456            if (c < min) {
457              min = c;
458              _in_arc = e;
459            }
460            else if (c >= 0) {
461              _candidates[i--] = _candidates[--_curr_length];
462            }
463          }
464          if (min < 0) return true;
465        }
466
467        // Major iteration: build a new candidate list
468        min = 0;
469        _curr_length = 0;
470        for (e = _next_arc; e < _search_arc_num; ++e) {
471          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
472          if (c < 0) {
473            _candidates[_curr_length++] = e;
474            if (c < min) {
475              min = c;
476              _in_arc = e;
477            }
478            if (_curr_length == _list_length) goto search_end;
479          }
480        }
481        for (e = 0; e < _next_arc; ++e) {
482          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
483          if (c < 0) {
484            _candidates[_curr_length++] = e;
485            if (c < min) {
486              min = c;
487              _in_arc = e;
488            }
489            if (_curr_length == _list_length) goto search_end;
490          }
491        }
492        if (_curr_length == 0) return false;
493     
494      search_end:       
495        _minor_count = 1;
496        _next_arc = e;
497        return true;
498      }
499
500    }; //class CandidateListPivotRule
501
502
503    // Implementation of the Altering Candidate List pivot rule
504    class AlteringListPivotRule
505    {
506    private:
507
508      // References to the NetworkSimplex class
509      const IntVector  &_source;
510      const IntVector  &_target;
511      const CostVector &_cost;
512      const IntVector  &_state;
513      const CostVector &_pi;
514      int &_in_arc;
515      int _search_arc_num;
516
517      // Pivot rule data
518      int _block_size, _head_length, _curr_length;
519      int _next_arc;
520      IntVector _candidates;
521      CostVector _cand_cost;
522
523      // Functor class to compare arcs during sort of the candidate list
524      class SortFunc
525      {
526      private:
527        const CostVector &_map;
528      public:
529        SortFunc(const CostVector &map) : _map(map) {}
530        bool operator()(int left, int right) {
531          return _map[left] > _map[right];
532        }
533      };
534
535      SortFunc _sort_func;
536
537    public:
538
539      // Constructor
540      AlteringListPivotRule(NetworkSimplex &ns) :
541        _source(ns._source), _target(ns._target),
542        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
543        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
544        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
545      {
546        // The main parameters of the pivot rule
547        const double BLOCK_SIZE_FACTOR = 1.0;
548        const int MIN_BLOCK_SIZE = 10;
549        const double HEAD_LENGTH_FACTOR = 0.1;
550        const int MIN_HEAD_LENGTH = 3;
551
552        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
553                                    std::sqrt(double(_search_arc_num))),
554                                MIN_BLOCK_SIZE );
555        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
556                                 MIN_HEAD_LENGTH );
557        _candidates.resize(_head_length + _block_size);
558        _curr_length = 0;
559      }
560
561      // Find next entering arc
562      bool findEnteringArc() {
563        // Check the current candidate list
564        int e;
565        for (int i = 0; i < _curr_length; ++i) {
566          e = _candidates[i];
567          _cand_cost[e] = _state[e] *
568            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
569          if (_cand_cost[e] >= 0) {
570            _candidates[i--] = _candidates[--_curr_length];
571          }
572        }
573
574        // Extend the list
575        int cnt = _block_size;
576        int limit = _head_length;
577
578        for (e = _next_arc; e < _search_arc_num; ++e) {
579          _cand_cost[e] = _state[e] *
580            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
581          if (_cand_cost[e] < 0) {
582            _candidates[_curr_length++] = e;
583          }
584          if (--cnt == 0) {
585            if (_curr_length > limit) goto search_end;
586            limit = 0;
587            cnt = _block_size;
588          }
589        }
590        for (e = 0; e < _next_arc; ++e) {
591          _cand_cost[e] = _state[e] *
592            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
593          if (_cand_cost[e] < 0) {
594            _candidates[_curr_length++] = e;
595          }
596          if (--cnt == 0) {
597            if (_curr_length > limit) goto search_end;
598            limit = 0;
599            cnt = _block_size;
600          }
601        }
602        if (_curr_length == 0) return false;
603       
604      search_end:
605
606        // Make heap of the candidate list (approximating a partial sort)
607        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
608                   _sort_func );
609
610        // Pop the first element of the heap
611        _in_arc = _candidates[0];
612        _next_arc = e;
613        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
614                  _sort_func );
615        _curr_length = std::min(_head_length, _curr_length - 1);
616        return true;
617      }
618
619    }; //class AlteringListPivotRule
620
621  public:
622
623    /// \brief Constructor.
624    ///
625    /// The constructor of the class.
626    ///
627    /// \param graph The digraph the algorithm runs on.
628    NetworkSimplex(const GR& graph) :
629      _graph(graph), _node_id(graph), _arc_id(graph),
630      INF(std::numeric_limits<Value>::has_infinity ?
631          std::numeric_limits<Value>::infinity() :
632          std::numeric_limits<Value>::max())
633    {
634      // Check the value types
635      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
636        "The flow type of NetworkSimplex must be signed");
637      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
638        "The cost type of NetworkSimplex must be signed");
639       
640      // Resize vectors
641      _node_num = countNodes(_graph);
642      _arc_num = countArcs(_graph);
643      int all_node_num = _node_num + 1;
644      int max_arc_num = _arc_num + 2 * _node_num;
645
646      _source.resize(max_arc_num);
647      _target.resize(max_arc_num);
648
649      _lower.resize(_arc_num);
650      _upper.resize(_arc_num);
651      _cap.resize(max_arc_num);
652      _cost.resize(max_arc_num);
653      _supply.resize(all_node_num);
654      _flow.resize(max_arc_num);
655      _pi.resize(all_node_num);
656
657      _parent.resize(all_node_num);
658      _pred.resize(all_node_num);
659      _forward.resize(all_node_num);
660      _thread.resize(all_node_num);
661      _rev_thread.resize(all_node_num);
662      _succ_num.resize(all_node_num);
663      _last_succ.resize(all_node_num);
664      _state.resize(max_arc_num);
665
666      // Copy the graph (store the arcs in a mixed order)
667      int i = 0;
668      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
669        _node_id[n] = i;
670      }
671      int k = std::max(int(std::sqrt(double(_arc_num))), 10);
672      i = 0;
673      for (ArcIt a(_graph); a != INVALID; ++a) {
674        _arc_id[a] = i;
675        _source[i] = _node_id[_graph.source(a)];
676        _target[i] = _node_id[_graph.target(a)];
677        if ((i += k) >= _arc_num) i = (i % k) + 1;
678      }
679     
680      // Initialize maps
681      for (int i = 0; i != _node_num; ++i) {
682        _supply[i] = 0;
683      }
684      for (int i = 0; i != _arc_num; ++i) {
685        _lower[i] = 0;
686        _upper[i] = INF;
687        _cost[i] = 1;
688      }
689      _have_lower = false;
690      _stype = GEQ;
691    }
692
693    /// \name Parameters
694    /// The parameters of the algorithm can be specified using these
695    /// functions.
696
697    /// @{
698
699    /// \brief Set the lower bounds on the arcs.
700    ///
701    /// This function sets the lower bounds on the arcs.
702    /// If it is not used before calling \ref run(), the lower bounds
703    /// will be set to zero on all arcs.
704    ///
705    /// \param map An arc map storing the lower bounds.
706    /// Its \c Value type must be convertible to the \c Value type
707    /// of the algorithm.
708    ///
709    /// \return <tt>(*this)</tt>
710    template <typename LowerMap>
711    NetworkSimplex& lowerMap(const LowerMap& map) {
712      _have_lower = true;
713      for (ArcIt a(_graph); a != INVALID; ++a) {
714        _lower[_arc_id[a]] = map[a];
715      }
716      return *this;
717    }
718
719    /// \brief Set the upper bounds (capacities) on the arcs.
720    ///
721    /// This function sets the upper bounds (capacities) on the arcs.
722    /// If it is not used before calling \ref run(), the upper bounds
723    /// will be set to \ref INF on all arcs (i.e. the flow value will be
724    /// unbounded from above on each arc).
725    ///
726    /// \param map An arc map storing the upper bounds.
727    /// Its \c Value type must be convertible to the \c Value type
728    /// of the algorithm.
729    ///
730    /// \return <tt>(*this)</tt>
731    template<typename UpperMap>
732    NetworkSimplex& upperMap(const UpperMap& map) {
733      for (ArcIt a(_graph); a != INVALID; ++a) {
734        _upper[_arc_id[a]] = map[a];
735      }
736      return *this;
737    }
738
739    /// \brief Set the costs of the arcs.
740    ///
741    /// This function sets the costs of the arcs.
742    /// If it is not used before calling \ref run(), the costs
743    /// will be set to \c 1 on all arcs.
744    ///
745    /// \param map An arc map storing the costs.
746    /// Its \c Value type must be convertible to the \c Cost type
747    /// of the algorithm.
748    ///
749    /// \return <tt>(*this)</tt>
750    template<typename CostMap>
751    NetworkSimplex& costMap(const CostMap& map) {
752      for (ArcIt a(_graph); a != INVALID; ++a) {
753        _cost[_arc_id[a]] = map[a];
754      }
755      return *this;
756    }
757
758    /// \brief Set the supply values of the nodes.
759    ///
760    /// This function sets the supply values of the nodes.
761    /// If neither this function nor \ref stSupply() is used before
762    /// calling \ref run(), the supply of each node will be set to zero.
763    /// (It makes sense only if non-zero lower bounds are given.)
764    ///
765    /// \param map A node map storing the supply values.
766    /// Its \c Value type must be convertible to the \c Value type
767    /// of the algorithm.
768    ///
769    /// \return <tt>(*this)</tt>
770    template<typename SupplyMap>
771    NetworkSimplex& supplyMap(const SupplyMap& map) {
772      for (NodeIt n(_graph); n != INVALID; ++n) {
773        _supply[_node_id[n]] = map[n];
774      }
775      return *this;
776    }
777
778    /// \brief Set single source and target nodes and a supply value.
779    ///
780    /// This function sets a single source node and a single target node
781    /// and the required flow value.
782    /// If neither this function nor \ref supplyMap() is used before
783    /// calling \ref run(), the supply of each node will be set to zero.
784    /// (It makes sense only if non-zero lower bounds are given.)
785    ///
786    /// Using this function has the same effect as using \ref supplyMap()
787    /// with such a map in which \c k is assigned to \c s, \c -k is
788    /// assigned to \c t and all other nodes have zero supply value.
789    ///
790    /// \param s The source node.
791    /// \param t The target node.
792    /// \param k The required amount of flow from node \c s to node \c t
793    /// (i.e. the supply of \c s and the demand of \c t).
794    ///
795    /// \return <tt>(*this)</tt>
796    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
797      for (int i = 0; i != _node_num; ++i) {
798        _supply[i] = 0;
799      }
800      _supply[_node_id[s]] =  k;
801      _supply[_node_id[t]] = -k;
802      return *this;
803    }
804   
805    /// \brief Set the type of the supply constraints.
806    ///
807    /// This function sets the type of the supply/demand constraints.
808    /// If it is not used before calling \ref run(), the \ref GEQ supply
809    /// type will be used.
810    ///
811    /// For more information see \ref SupplyType.
812    ///
813    /// \return <tt>(*this)</tt>
814    NetworkSimplex& supplyType(SupplyType supply_type) {
815      _stype = supply_type;
816      return *this;
817    }
818
819    /// @}
820
821    /// \name Execution Control
822    /// The algorithm can be executed using \ref run().
823
824    /// @{
825
826    /// \brief Run the algorithm.
827    ///
828    /// This function runs the algorithm.
829    /// The paramters can be specified using functions \ref lowerMap(),
830    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
831    /// \ref supplyType().
832    /// For example,
833    /// \code
834    ///   NetworkSimplex<ListDigraph> ns(graph);
835    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
836    ///     .supplyMap(sup).run();
837    /// \endcode
838    ///
839    /// This function can be called more than once. All the parameters
840    /// that have been given are kept for the next call, unless
841    /// \ref reset() is called, thus only the modified parameters
842    /// have to be set again. See \ref reset() for examples.
843    /// However the underlying digraph must not be modified after this
844    /// class have been constructed, since it copies and extends the graph.
845    ///
846    /// \param pivot_rule The pivot rule that will be used during the
847    /// algorithm. For more information see \ref PivotRule.
848    ///
849    /// \return \c INFEASIBLE if no feasible flow exists,
850    /// \n \c OPTIMAL if the problem has optimal solution
851    /// (i.e. it is feasible and bounded), and the algorithm has found
852    /// optimal flow and node potentials (primal and dual solutions),
853    /// \n \c UNBOUNDED if the objective function of the problem is
854    /// unbounded, i.e. there is a directed cycle having negative total
855    /// cost and infinite upper bound.
856    ///
857    /// \see ProblemType, PivotRule
858    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
859      if (!init()) return INFEASIBLE;
860      return start(pivot_rule);
861    }
862
863    /// \brief Reset all the parameters that have been given before.
864    ///
865    /// This function resets all the paramaters that have been given
866    /// before using functions \ref lowerMap(), \ref upperMap(),
867    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
868    ///
869    /// It is useful for multiple run() calls. If this function is not
870    /// used, all the parameters given before are kept for the next
871    /// \ref run() call.
872    /// However the underlying digraph must not be modified after this
873    /// class have been constructed, since it copies and extends the graph.
874    ///
875    /// For example,
876    /// \code
877    ///   NetworkSimplex<ListDigraph> ns(graph);
878    ///
879    ///   // First run
880    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
881    ///     .supplyMap(sup).run();
882    ///
883    ///   // Run again with modified cost map (reset() is not called,
884    ///   // so only the cost map have to be set again)
885    ///   cost[e] += 100;
886    ///   ns.costMap(cost).run();
887    ///
888    ///   // Run again from scratch using reset()
889    ///   // (the lower bounds will be set to zero on all arcs)
890    ///   ns.reset();
891    ///   ns.upperMap(capacity).costMap(cost)
892    ///     .supplyMap(sup).run();
893    /// \endcode
894    ///
895    /// \return <tt>(*this)</tt>
896    NetworkSimplex& reset() {
897      for (int i = 0; i != _node_num; ++i) {
898        _supply[i] = 0;
899      }
900      for (int i = 0; i != _arc_num; ++i) {
901        _lower[i] = 0;
902        _upper[i] = INF;
903        _cost[i] = 1;
904      }
905      _have_lower = false;
906      _stype = GEQ;
907      return *this;
908    }
909
910    /// @}
911
912    /// \name Query Functions
913    /// The results of the algorithm can be obtained using these
914    /// functions.\n
915    /// The \ref run() function must be called before using them.
916
917    /// @{
918
919    /// \brief Return the total cost of the found flow.
920    ///
921    /// This function returns the total cost of the found flow.
922    /// Its complexity is O(e).
923    ///
924    /// \note The return type of the function can be specified as a
925    /// template parameter. For example,
926    /// \code
927    ///   ns.totalCost<double>();
928    /// \endcode
929    /// It is useful if the total cost cannot be stored in the \c Cost
930    /// type of the algorithm, which is the default return type of the
931    /// function.
932    ///
933    /// \pre \ref run() must be called before using this function.
934    template <typename Number>
935    Number totalCost() const {
936      Number c = 0;
937      for (ArcIt a(_graph); a != INVALID; ++a) {
938        int i = _arc_id[a];
939        c += Number(_flow[i]) * Number(_cost[i]);
940      }
941      return c;
942    }
943
944#ifndef DOXYGEN
945    Cost totalCost() const {
946      return totalCost<Cost>();
947    }
948#endif
949
950    /// \brief Return the flow on the given arc.
951    ///
952    /// This function returns the flow on the given arc.
953    ///
954    /// \pre \ref run() must be called before using this function.
955    Value flow(const Arc& a) const {
956      return _flow[_arc_id[a]];
957    }
958
959    /// \brief Return the flow map (the primal solution).
960    ///
961    /// This function copies the flow value on each arc into the given
962    /// map. The \c Value type of the algorithm must be convertible to
963    /// the \c Value type of the map.
964    ///
965    /// \pre \ref run() must be called before using this function.
966    template <typename FlowMap>
967    void flowMap(FlowMap &map) const {
968      for (ArcIt a(_graph); a != INVALID; ++a) {
969        map.set(a, _flow[_arc_id[a]]);
970      }
971    }
972
973    /// \brief Return the potential (dual value) of the given node.
974    ///
975    /// This function returns the potential (dual value) of the
976    /// given node.
977    ///
978    /// \pre \ref run() must be called before using this function.
979    Cost potential(const Node& n) const {
980      return _pi[_node_id[n]];
981    }
982
983    /// \brief Return the potential map (the dual solution).
984    ///
985    /// This function copies the potential (dual value) of each node
986    /// into the given map.
987    /// The \c Cost type of the algorithm must be convertible to the
988    /// \c Value type of the map.
989    ///
990    /// \pre \ref run() must be called before using this function.
991    template <typename PotentialMap>
992    void potentialMap(PotentialMap &map) const {
993      for (NodeIt n(_graph); n != INVALID; ++n) {
994        map.set(n, _pi[_node_id[n]]);
995      }
996    }
997
998    /// @}
999
1000  private:
1001
1002    // Initialize internal data structures
1003    bool init() {
1004      if (_node_num == 0) return false;
1005
1006      // Check the sum of supply values
1007      _sum_supply = 0;
1008      for (int i = 0; i != _node_num; ++i) {
1009        _sum_supply += _supply[i];
1010      }
1011      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1012             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1013
1014      // Remove non-zero lower bounds
1015      if (_have_lower) {
1016        for (int i = 0; i != _arc_num; ++i) {
1017          Value c = _lower[i];
1018          if (c >= 0) {
1019            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1020          } else {
1021            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1022          }
1023          _supply[_source[i]] -= c;
1024          _supply[_target[i]] += c;
1025        }
1026      } else {
1027        for (int i = 0; i != _arc_num; ++i) {
1028          _cap[i] = _upper[i];
1029        }
1030      }
1031
1032      // Initialize artifical cost
1033      Cost ART_COST;
1034      if (std::numeric_limits<Cost>::is_exact) {
1035        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1036      } else {
1037        ART_COST = std::numeric_limits<Cost>::min();
1038        for (int i = 0; i != _arc_num; ++i) {
1039          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1040        }
1041        ART_COST = (ART_COST + 1) * _node_num;
1042      }
1043
1044      // Initialize arc maps
1045      for (int i = 0; i != _arc_num; ++i) {
1046        _flow[i] = 0;
1047        _state[i] = STATE_LOWER;
1048      }
1049     
1050      // Set data for the artificial root node
1051      _root = _node_num;
1052      _parent[_root] = -1;
1053      _pred[_root] = -1;
1054      _thread[_root] = 0;
1055      _rev_thread[0] = _root;
1056      _succ_num[_root] = _node_num + 1;
1057      _last_succ[_root] = _root - 1;
1058      _supply[_root] = -_sum_supply;
1059      _pi[_root] = 0;
1060
1061      // Add artificial arcs and initialize the spanning tree data structure
1062      if (_sum_supply == 0) {
1063        // EQ supply constraints
1064        _search_arc_num = _arc_num;
1065        _all_arc_num = _arc_num + _node_num;
1066        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1067          _parent[u] = _root;
1068          _pred[u] = e;
1069          _thread[u] = u + 1;
1070          _rev_thread[u + 1] = u;
1071          _succ_num[u] = 1;
1072          _last_succ[u] = u;
1073          _cap[e] = INF;
1074          _state[e] = STATE_TREE;
1075          if (_supply[u] >= 0) {
1076            _forward[u] = true;
1077            _pi[u] = 0;
1078            _source[e] = u;
1079            _target[e] = _root;
1080            _flow[e] = _supply[u];
1081            _cost[e] = 0;
1082          } else {
1083            _forward[u] = false;
1084            _pi[u] = ART_COST;
1085            _source[e] = _root;
1086            _target[e] = u;
1087            _flow[e] = -_supply[u];
1088            _cost[e] = ART_COST;
1089          }
1090        }
1091      }
1092      else if (_sum_supply > 0) {
1093        // LEQ supply constraints
1094        _search_arc_num = _arc_num + _node_num;
1095        int f = _arc_num + _node_num;
1096        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1097          _parent[u] = _root;
1098          _thread[u] = u + 1;
1099          _rev_thread[u + 1] = u;
1100          _succ_num[u] = 1;
1101          _last_succ[u] = u;
1102          if (_supply[u] >= 0) {
1103            _forward[u] = true;
1104            _pi[u] = 0;
1105            _pred[u] = e;
1106            _source[e] = u;
1107            _target[e] = _root;
1108            _cap[e] = INF;
1109            _flow[e] = _supply[u];
1110            _cost[e] = 0;
1111            _state[e] = STATE_TREE;
1112          } else {
1113            _forward[u] = false;
1114            _pi[u] = ART_COST;
1115            _pred[u] = f;
1116            _source[f] = _root;
1117            _target[f] = u;
1118            _cap[f] = INF;
1119            _flow[f] = -_supply[u];
1120            _cost[f] = ART_COST;
1121            _state[f] = STATE_TREE;
1122            _source[e] = u;
1123            _target[e] = _root;
1124            _cap[e] = INF;
1125            _flow[e] = 0;
1126            _cost[e] = 0;
1127            _state[e] = STATE_LOWER;
1128            ++f;
1129          }
1130        }
1131        _all_arc_num = f;
1132      }
1133      else {
1134        // GEQ supply constraints
1135        _search_arc_num = _arc_num + _node_num;
1136        int f = _arc_num + _node_num;
1137        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1138          _parent[u] = _root;
1139          _thread[u] = u + 1;
1140          _rev_thread[u + 1] = u;
1141          _succ_num[u] = 1;
1142          _last_succ[u] = u;
1143          if (_supply[u] <= 0) {
1144            _forward[u] = false;
1145            _pi[u] = 0;
1146            _pred[u] = e;
1147            _source[e] = _root;
1148            _target[e] = u;
1149            _cap[e] = INF;
1150            _flow[e] = -_supply[u];
1151            _cost[e] = 0;
1152            _state[e] = STATE_TREE;
1153          } else {
1154            _forward[u] = true;
1155            _pi[u] = -ART_COST;
1156            _pred[u] = f;
1157            _source[f] = u;
1158            _target[f] = _root;
1159            _cap[f] = INF;
1160            _flow[f] = _supply[u];
1161            _state[f] = STATE_TREE;
1162            _cost[f] = ART_COST;
1163            _source[e] = _root;
1164            _target[e] = u;
1165            _cap[e] = INF;
1166            _flow[e] = 0;
1167            _cost[e] = 0;
1168            _state[e] = STATE_LOWER;
1169            ++f;
1170          }
1171        }
1172        _all_arc_num = f;
1173      }
1174
1175      return true;
1176    }
1177
1178    // Find the join node
1179    void findJoinNode() {
1180      int u = _source[in_arc];
1181      int v = _target[in_arc];
1182      while (u != v) {
1183        if (_succ_num[u] < _succ_num[v]) {
1184          u = _parent[u];
1185        } else {
1186          v = _parent[v];
1187        }
1188      }
1189      join = u;
1190    }
1191
1192    // Find the leaving arc of the cycle and returns true if the
1193    // leaving arc is not the same as the entering arc
1194    bool findLeavingArc() {
1195      // Initialize first and second nodes according to the direction
1196      // of the cycle
1197      if (_state[in_arc] == STATE_LOWER) {
1198        first  = _source[in_arc];
1199        second = _target[in_arc];
1200      } else {
1201        first  = _target[in_arc];
1202        second = _source[in_arc];
1203      }
1204      delta = _cap[in_arc];
1205      int result = 0;
1206      Value d;
1207      int e;
1208
1209      // Search the cycle along the path form the first node to the root
1210      for (int u = first; u != join; u = _parent[u]) {
1211        e = _pred[u];
1212        d = _forward[u] ?
1213          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1214        if (d < delta) {
1215          delta = d;
1216          u_out = u;
1217          result = 1;
1218        }
1219      }
1220      // Search the cycle along the path form the second node to the root
1221      for (int u = second; u != join; u = _parent[u]) {
1222        e = _pred[u];
1223        d = _forward[u] ?
1224          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1225        if (d <= delta) {
1226          delta = d;
1227          u_out = u;
1228          result = 2;
1229        }
1230      }
1231
1232      if (result == 1) {
1233        u_in = first;
1234        v_in = second;
1235      } else {
1236        u_in = second;
1237        v_in = first;
1238      }
1239      return result != 0;
1240    }
1241
1242    // Change _flow and _state vectors
1243    void changeFlow(bool change) {
1244      // Augment along the cycle
1245      if (delta > 0) {
1246        Value val = _state[in_arc] * delta;
1247        _flow[in_arc] += val;
1248        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1249          _flow[_pred[u]] += _forward[u] ? -val : val;
1250        }
1251        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1252          _flow[_pred[u]] += _forward[u] ? val : -val;
1253        }
1254      }
1255      // Update the state of the entering and leaving arcs
1256      if (change) {
1257        _state[in_arc] = STATE_TREE;
1258        _state[_pred[u_out]] =
1259          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1260      } else {
1261        _state[in_arc] = -_state[in_arc];
1262      }
1263    }
1264
1265    // Update the tree structure
1266    void updateTreeStructure() {
1267      int u, w;
1268      int old_rev_thread = _rev_thread[u_out];
1269      int old_succ_num = _succ_num[u_out];
1270      int old_last_succ = _last_succ[u_out];
1271      v_out = _parent[u_out];
1272
1273      u = _last_succ[u_in];  // the last successor of u_in
1274      right = _thread[u];    // the node after it
1275
1276      // Handle the case when old_rev_thread equals to v_in
1277      // (it also means that join and v_out coincide)
1278      if (old_rev_thread == v_in) {
1279        last = _thread[_last_succ[u_out]];
1280      } else {
1281        last = _thread[v_in];
1282      }
1283
1284      // Update _thread and _parent along the stem nodes (i.e. the nodes
1285      // between u_in and u_out, whose parent have to be changed)
1286      _thread[v_in] = stem = u_in;
1287      _dirty_revs.clear();
1288      _dirty_revs.push_back(v_in);
1289      par_stem = v_in;
1290      while (stem != u_out) {
1291        // Insert the next stem node into the thread list
1292        new_stem = _parent[stem];
1293        _thread[u] = new_stem;
1294        _dirty_revs.push_back(u);
1295
1296        // Remove the subtree of stem from the thread list
1297        w = _rev_thread[stem];
1298        _thread[w] = right;
1299        _rev_thread[right] = w;
1300
1301        // Change the parent node and shift stem nodes
1302        _parent[stem] = par_stem;
1303        par_stem = stem;
1304        stem = new_stem;
1305
1306        // Update u and right
1307        u = _last_succ[stem] == _last_succ[par_stem] ?
1308          _rev_thread[par_stem] : _last_succ[stem];
1309        right = _thread[u];
1310      }
1311      _parent[u_out] = par_stem;
1312      _thread[u] = last;
1313      _rev_thread[last] = u;
1314      _last_succ[u_out] = u;
1315
1316      // Remove the subtree of u_out from the thread list except for
1317      // the case when old_rev_thread equals to v_in
1318      // (it also means that join and v_out coincide)
1319      if (old_rev_thread != v_in) {
1320        _thread[old_rev_thread] = right;
1321        _rev_thread[right] = old_rev_thread;
1322      }
1323
1324      // Update _rev_thread using the new _thread values
1325      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1326        u = _dirty_revs[i];
1327        _rev_thread[_thread[u]] = u;
1328      }
1329
1330      // Update _pred, _forward, _last_succ and _succ_num for the
1331      // stem nodes from u_out to u_in
1332      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1333      u = u_out;
1334      while (u != u_in) {
1335        w = _parent[u];
1336        _pred[u] = _pred[w];
1337        _forward[u] = !_forward[w];
1338        tmp_sc += _succ_num[u] - _succ_num[w];
1339        _succ_num[u] = tmp_sc;
1340        _last_succ[w] = tmp_ls;
1341        u = w;
1342      }
1343      _pred[u_in] = in_arc;
1344      _forward[u_in] = (u_in == _source[in_arc]);
1345      _succ_num[u_in] = old_succ_num;
1346
1347      // Set limits for updating _last_succ form v_in and v_out
1348      // towards the root
1349      int up_limit_in = -1;
1350      int up_limit_out = -1;
1351      if (_last_succ[join] == v_in) {
1352        up_limit_out = join;
1353      } else {
1354        up_limit_in = join;
1355      }
1356
1357      // Update _last_succ from v_in towards the root
1358      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1359           u = _parent[u]) {
1360        _last_succ[u] = _last_succ[u_out];
1361      }
1362      // Update _last_succ from v_out towards the root
1363      if (join != old_rev_thread && v_in != old_rev_thread) {
1364        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1365             u = _parent[u]) {
1366          _last_succ[u] = old_rev_thread;
1367        }
1368      } else {
1369        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1370             u = _parent[u]) {
1371          _last_succ[u] = _last_succ[u_out];
1372        }
1373      }
1374
1375      // Update _succ_num from v_in to join
1376      for (u = v_in; u != join; u = _parent[u]) {
1377        _succ_num[u] += old_succ_num;
1378      }
1379      // Update _succ_num from v_out to join
1380      for (u = v_out; u != join; u = _parent[u]) {
1381        _succ_num[u] -= old_succ_num;
1382      }
1383    }
1384
1385    // Update potentials
1386    void updatePotential() {
1387      Cost sigma = _forward[u_in] ?
1388        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1389        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1390      // Update potentials in the subtree, which has been moved
1391      int end = _thread[_last_succ[u_in]];
1392      for (int u = u_in; u != end; u = _thread[u]) {
1393        _pi[u] += sigma;
1394      }
1395    }
1396
1397    // Execute the algorithm
1398    ProblemType start(PivotRule pivot_rule) {
1399      // Select the pivot rule implementation
1400      switch (pivot_rule) {
1401        case FIRST_ELIGIBLE:
1402          return start<FirstEligiblePivotRule>();
1403        case BEST_ELIGIBLE:
1404          return start<BestEligiblePivotRule>();
1405        case BLOCK_SEARCH:
1406          return start<BlockSearchPivotRule>();
1407        case CANDIDATE_LIST:
1408          return start<CandidateListPivotRule>();
1409        case ALTERING_LIST:
1410          return start<AlteringListPivotRule>();
1411      }
1412      return INFEASIBLE; // avoid warning
1413    }
1414
1415    template <typename PivotRuleImpl>
1416    ProblemType start() {
1417      PivotRuleImpl pivot(*this);
1418
1419      // Execute the Network Simplex algorithm
1420      while (pivot.findEnteringArc()) {
1421        findJoinNode();
1422        bool change = findLeavingArc();
1423        if (delta >= INF) return UNBOUNDED;
1424        changeFlow(change);
1425        if (change) {
1426          updateTreeStructure();
1427          updatePotential();
1428        }
1429      }
1430     
1431      // Check feasibility
1432      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1433        if (_flow[e] != 0) return INFEASIBLE;
1434      }
1435
1436      // Transform the solution and the supply map to the original form
1437      if (_have_lower) {
1438        for (int i = 0; i != _arc_num; ++i) {
1439          Value c = _lower[i];
1440          if (c != 0) {
1441            _flow[i] += c;
1442            _supply[_source[i]] += c;
1443            _supply[_target[i]] -= c;
1444          }
1445        }
1446      }
1447     
1448      // Shift potentials to meet the requirements of the GEQ/LEQ type
1449      // optimality conditions
1450      if (_sum_supply == 0) {
1451        if (_stype == GEQ) {
1452          Cost max_pot = std::numeric_limits<Cost>::min();
1453          for (int i = 0; i != _node_num; ++i) {
1454            if (_pi[i] > max_pot) max_pot = _pi[i];
1455          }
1456          if (max_pot > 0) {
1457            for (int i = 0; i != _node_num; ++i)
1458              _pi[i] -= max_pot;
1459          }
1460        } else {
1461          Cost min_pot = std::numeric_limits<Cost>::max();
1462          for (int i = 0; i != _node_num; ++i) {
1463            if (_pi[i] < min_pot) min_pot = _pi[i];
1464          }
1465          if (min_pot < 0) {
1466            for (int i = 0; i != _node_num; ++i)
1467              _pi[i] -= min_pot;
1468          }
1469        }
1470      }
1471
1472      return OPTIMAL;
1473    }
1474
1475  }; //class NetworkSimplex
1476
1477  ///@}
1478
1479} //namespace lemon
1480
1481#endif //LEMON_NETWORK_SIMPLEX_H
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