COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 710:8b0df68370a4

Last change on this file since 710:8b0df68370a4 was 710:8b0df68370a4, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Fix the GEQ/LEQ handling in NetworkSimplex? + improve doc (#291)

  • Fix the optimality conditions for the GEQ/LEQ form.
  • Fix the initialization of the algortihm. It ensures correct solutions and it is much faster for the inequality forms.
  • Fix the pivot rules to search all the arcs that have to be allowed to get in the basis.
  • Better block size for the Block Search pivot rule.
  • Improve documentation of the problem and move it to a separate page.
File size: 45.4 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, min_arc = _next_arc;
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            min_arc = e;
373          }
374          if (--cnt == 0) {
375            if (min < 0) break;
376            cnt = _block_size;
377          }
378        }
379        if (min == 0 || cnt > 0) {
380          for (e = 0; e < _next_arc; ++e) {
381            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
382            if (c < min) {
383              min = c;
384              min_arc = e;
385            }
386            if (--cnt == 0) {
387              if (min < 0) break;
388              cnt = _block_size;
389            }
390          }
391        }
392        if (min >= 0) return false;
393        _in_arc = min_arc;
394        _next_arc = e;
395        return true;
396      }
397
398    }; //class BlockSearchPivotRule
399
400
401    // Implementation of the Candidate List pivot rule
402    class CandidateListPivotRule
403    {
404    private:
405
406      // References to the NetworkSimplex class
407      const IntVector  &_source;
408      const IntVector  &_target;
409      const CostVector &_cost;
410      const IntVector  &_state;
411      const CostVector &_pi;
412      int &_in_arc;
413      int _search_arc_num;
414
415      // Pivot rule data
416      IntVector _candidates;
417      int _list_length, _minor_limit;
418      int _curr_length, _minor_count;
419      int _next_arc;
420
421    public:
422
423      /// Constructor
424      CandidateListPivotRule(NetworkSimplex &ns) :
425        _source(ns._source), _target(ns._target),
426        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
427        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
428        _next_arc(0)
429      {
430        // The main parameters of the pivot rule
431        const double LIST_LENGTH_FACTOR = 1.0;
432        const int MIN_LIST_LENGTH = 10;
433        const double MINOR_LIMIT_FACTOR = 0.1;
434        const int MIN_MINOR_LIMIT = 3;
435
436        _list_length = std::max( int(LIST_LENGTH_FACTOR *
437                                     std::sqrt(double(_search_arc_num))),
438                                 MIN_LIST_LENGTH );
439        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
440                                 MIN_MINOR_LIMIT );
441        _curr_length = _minor_count = 0;
442        _candidates.resize(_list_length);
443      }
444
445      /// Find next entering arc
446      bool findEnteringArc() {
447        Cost min, c;
448        int e, min_arc = _next_arc;
449        if (_curr_length > 0 && _minor_count < _minor_limit) {
450          // Minor iteration: select the best eligible arc from the
451          // current candidate list
452          ++_minor_count;
453          min = 0;
454          for (int i = 0; i < _curr_length; ++i) {
455            e = _candidates[i];
456            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
457            if (c < min) {
458              min = c;
459              min_arc = e;
460            }
461            if (c >= 0) {
462              _candidates[i--] = _candidates[--_curr_length];
463            }
464          }
465          if (min < 0) {
466            _in_arc = min_arc;
467            return true;
468          }
469        }
470
471        // Major iteration: build a new candidate list
472        min = 0;
473        _curr_length = 0;
474        for (e = _next_arc; e < _search_arc_num; ++e) {
475          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
476          if (c < 0) {
477            _candidates[_curr_length++] = e;
478            if (c < min) {
479              min = c;
480              min_arc = e;
481            }
482            if (_curr_length == _list_length) break;
483          }
484        }
485        if (_curr_length < _list_length) {
486          for (e = 0; e < _next_arc; ++e) {
487            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
488            if (c < 0) {
489              _candidates[_curr_length++] = e;
490              if (c < min) {
491                min = c;
492                min_arc = e;
493              }
494              if (_curr_length == _list_length) break;
495            }
496          }
497        }
498        if (_curr_length == 0) return false;
499        _minor_count = 1;
500        _in_arc = min_arc;
501        _next_arc = e;
502        return true;
503      }
504
505    }; //class CandidateListPivotRule
506
507
508    // Implementation of the Altering Candidate List pivot rule
509    class AlteringListPivotRule
510    {
511    private:
512
513      // References to the NetworkSimplex class
514      const IntVector  &_source;
515      const IntVector  &_target;
516      const CostVector &_cost;
517      const IntVector  &_state;
518      const CostVector &_pi;
519      int &_in_arc;
520      int _search_arc_num;
521
522      // Pivot rule data
523      int _block_size, _head_length, _curr_length;
524      int _next_arc;
525      IntVector _candidates;
526      CostVector _cand_cost;
527
528      // Functor class to compare arcs during sort of the candidate list
529      class SortFunc
530      {
531      private:
532        const CostVector &_map;
533      public:
534        SortFunc(const CostVector &map) : _map(map) {}
535        bool operator()(int left, int right) {
536          return _map[left] > _map[right];
537        }
538      };
539
540      SortFunc _sort_func;
541
542    public:
543
544      // Constructor
545      AlteringListPivotRule(NetworkSimplex &ns) :
546        _source(ns._source), _target(ns._target),
547        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
548        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
549        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
550      {
551        // The main parameters of the pivot rule
552        const double BLOCK_SIZE_FACTOR = 1.5;
553        const int MIN_BLOCK_SIZE = 10;
554        const double HEAD_LENGTH_FACTOR = 0.1;
555        const int MIN_HEAD_LENGTH = 3;
556
557        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
558                                    std::sqrt(double(_search_arc_num))),
559                                MIN_BLOCK_SIZE );
560        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
561                                 MIN_HEAD_LENGTH );
562        _candidates.resize(_head_length + _block_size);
563        _curr_length = 0;
564      }
565
566      // Find next entering arc
567      bool findEnteringArc() {
568        // Check the current candidate list
569        int e;
570        for (int i = 0; i < _curr_length; ++i) {
571          e = _candidates[i];
572          _cand_cost[e] = _state[e] *
573            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
574          if (_cand_cost[e] >= 0) {
575            _candidates[i--] = _candidates[--_curr_length];
576          }
577        }
578
579        // Extend the list
580        int cnt = _block_size;
581        int last_arc = 0;
582        int limit = _head_length;
583
584        for (int e = _next_arc; e < _search_arc_num; ++e) {
585          _cand_cost[e] = _state[e] *
586            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
587          if (_cand_cost[e] < 0) {
588            _candidates[_curr_length++] = e;
589            last_arc = e;
590          }
591          if (--cnt == 0) {
592            if (_curr_length > limit) break;
593            limit = 0;
594            cnt = _block_size;
595          }
596        }
597        if (_curr_length <= limit) {
598          for (int e = 0; e < _next_arc; ++e) {
599            _cand_cost[e] = _state[e] *
600              (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
601            if (_cand_cost[e] < 0) {
602              _candidates[_curr_length++] = e;
603              last_arc = e;
604            }
605            if (--cnt == 0) {
606              if (_curr_length > limit) break;
607              limit = 0;
608              cnt = _block_size;
609            }
610          }
611        }
612        if (_curr_length == 0) return false;
613        _next_arc = last_arc + 1;
614
615        // Make heap of the candidate list (approximating a partial sort)
616        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
617                   _sort_func );
618
619        // Pop the first element of the heap
620        _in_arc = _candidates[0];
621        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
622                  _sort_func );
623        _curr_length = std::min(_head_length, _curr_length - 1);
624        return true;
625      }
626
627    }; //class AlteringListPivotRule
628
629  public:
630
631    /// \brief Constructor.
632    ///
633    /// The constructor of the class.
634    ///
635    /// \param graph The digraph the algorithm runs on.
636    NetworkSimplex(const GR& graph) :
637      _graph(graph), _node_id(graph), _arc_id(graph),
638      INF(std::numeric_limits<Value>::has_infinity ?
639          std::numeric_limits<Value>::infinity() :
640          std::numeric_limits<Value>::max())
641    {
642      // Check the value types
643      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
644        "The flow type of NetworkSimplex must be signed");
645      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
646        "The cost type of NetworkSimplex must be signed");
647       
648      // Resize vectors
649      _node_num = countNodes(_graph);
650      _arc_num = countArcs(_graph);
651      int all_node_num = _node_num + 1;
652      int max_arc_num = _arc_num + 2 * _node_num;
653
654      _source.resize(max_arc_num);
655      _target.resize(max_arc_num);
656
657      _lower.resize(_arc_num);
658      _upper.resize(_arc_num);
659      _cap.resize(max_arc_num);
660      _cost.resize(max_arc_num);
661      _supply.resize(all_node_num);
662      _flow.resize(max_arc_num);
663      _pi.resize(all_node_num);
664
665      _parent.resize(all_node_num);
666      _pred.resize(all_node_num);
667      _forward.resize(all_node_num);
668      _thread.resize(all_node_num);
669      _rev_thread.resize(all_node_num);
670      _succ_num.resize(all_node_num);
671      _last_succ.resize(all_node_num);
672      _state.resize(max_arc_num);
673
674      // Copy the graph (store the arcs in a mixed order)
675      int i = 0;
676      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
677        _node_id[n] = i;
678      }
679      int k = std::max(int(std::sqrt(double(_arc_num))), 10);
680      i = 0;
681      for (ArcIt a(_graph); a != INVALID; ++a) {
682        _arc_id[a] = i;
683        _source[i] = _node_id[_graph.source(a)];
684        _target[i] = _node_id[_graph.target(a)];
685        if ((i += k) >= _arc_num) i = (i % k) + 1;
686      }
687     
688      // Initialize maps
689      for (int i = 0; i != _node_num; ++i) {
690        _supply[i] = 0;
691      }
692      for (int i = 0; i != _arc_num; ++i) {
693        _lower[i] = 0;
694        _upper[i] = INF;
695        _cost[i] = 1;
696      }
697      _have_lower = false;
698      _stype = GEQ;
699    }
700
701    /// \name Parameters
702    /// The parameters of the algorithm can be specified using these
703    /// functions.
704
705    /// @{
706
707    /// \brief Set the lower bounds on the arcs.
708    ///
709    /// This function sets the lower bounds on the arcs.
710    /// If it is not used before calling \ref run(), the lower bounds
711    /// will be set to zero on all arcs.
712    ///
713    /// \param map An arc map storing the lower bounds.
714    /// Its \c Value type must be convertible to the \c Value type
715    /// of the algorithm.
716    ///
717    /// \return <tt>(*this)</tt>
718    template <typename LowerMap>
719    NetworkSimplex& lowerMap(const LowerMap& map) {
720      _have_lower = true;
721      for (ArcIt a(_graph); a != INVALID; ++a) {
722        _lower[_arc_id[a]] = map[a];
723      }
724      return *this;
725    }
726
727    /// \brief Set the upper bounds (capacities) on the arcs.
728    ///
729    /// This function sets the upper bounds (capacities) on the arcs.
730    /// If it is not used before calling \ref run(), the upper bounds
731    /// will be set to \ref INF on all arcs (i.e. the flow value will be
732    /// unbounded from above on each arc).
733    ///
734    /// \param map An arc map storing the upper bounds.
735    /// Its \c Value type must be convertible to the \c Value type
736    /// of the algorithm.
737    ///
738    /// \return <tt>(*this)</tt>
739    template<typename UpperMap>
740    NetworkSimplex& upperMap(const UpperMap& map) {
741      for (ArcIt a(_graph); a != INVALID; ++a) {
742        _upper[_arc_id[a]] = map[a];
743      }
744      return *this;
745    }
746
747    /// \brief Set the costs of the arcs.
748    ///
749    /// This function sets the costs of the arcs.
750    /// If it is not used before calling \ref run(), the costs
751    /// will be set to \c 1 on all arcs.
752    ///
753    /// \param map An arc map storing the costs.
754    /// Its \c Value type must be convertible to the \c Cost type
755    /// of the algorithm.
756    ///
757    /// \return <tt>(*this)</tt>
758    template<typename CostMap>
759    NetworkSimplex& costMap(const CostMap& map) {
760      for (ArcIt a(_graph); a != INVALID; ++a) {
761        _cost[_arc_id[a]] = map[a];
762      }
763      return *this;
764    }
765
766    /// \brief Set the supply values of the nodes.
767    ///
768    /// This function sets the supply values of the nodes.
769    /// If neither this function nor \ref stSupply() is used before
770    /// calling \ref run(), the supply of each node will be set to zero.
771    /// (It makes sense only if non-zero lower bounds are given.)
772    ///
773    /// \param map A node map storing the supply values.
774    /// Its \c Value type must be convertible to the \c Value type
775    /// of the algorithm.
776    ///
777    /// \return <tt>(*this)</tt>
778    template<typename SupplyMap>
779    NetworkSimplex& supplyMap(const SupplyMap& map) {
780      for (NodeIt n(_graph); n != INVALID; ++n) {
781        _supply[_node_id[n]] = map[n];
782      }
783      return *this;
784    }
785
786    /// \brief Set single source and target nodes and a supply value.
787    ///
788    /// This function sets a single source node and a single target node
789    /// and the required flow value.
790    /// If neither this function nor \ref supplyMap() is used before
791    /// calling \ref run(), the supply of each node will be set to zero.
792    /// (It makes sense only if non-zero lower bounds are given.)
793    ///
794    /// Using this function has the same effect as using \ref supplyMap()
795    /// with such a map in which \c k is assigned to \c s, \c -k is
796    /// assigned to \c t and all other nodes have zero supply value.
797    ///
798    /// \param s The source node.
799    /// \param t The target node.
800    /// \param k The required amount of flow from node \c s to node \c t
801    /// (i.e. the supply of \c s and the demand of \c t).
802    ///
803    /// \return <tt>(*this)</tt>
804    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
805      for (int i = 0; i != _node_num; ++i) {
806        _supply[i] = 0;
807      }
808      _supply[_node_id[s]] =  k;
809      _supply[_node_id[t]] = -k;
810      return *this;
811    }
812   
813    /// \brief Set the type of the supply constraints.
814    ///
815    /// This function sets the type of the supply/demand constraints.
816    /// If it is not used before calling \ref run(), the \ref GEQ supply
817    /// type will be used.
818    ///
819    /// For more information see \ref SupplyType.
820    ///
821    /// \return <tt>(*this)</tt>
822    NetworkSimplex& supplyType(SupplyType supply_type) {
823      _stype = supply_type;
824      return *this;
825    }
826
827    /// @}
828
829    /// \name Execution Control
830    /// The algorithm can be executed using \ref run().
831
832    /// @{
833
834    /// \brief Run the algorithm.
835    ///
836    /// This function runs the algorithm.
837    /// The paramters can be specified using functions \ref lowerMap(),
838    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
839    /// \ref supplyType().
840    /// For example,
841    /// \code
842    ///   NetworkSimplex<ListDigraph> ns(graph);
843    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
844    ///     .supplyMap(sup).run();
845    /// \endcode
846    ///
847    /// This function can be called more than once. All the parameters
848    /// that have been given are kept for the next call, unless
849    /// \ref reset() is called, thus only the modified parameters
850    /// have to be set again. See \ref reset() for examples.
851    /// However the underlying digraph must not be modified after this
852    /// class have been constructed, since it copies and extends the graph.
853    ///
854    /// \param pivot_rule The pivot rule that will be used during the
855    /// algorithm. For more information see \ref PivotRule.
856    ///
857    /// \return \c INFEASIBLE if no feasible flow exists,
858    /// \n \c OPTIMAL if the problem has optimal solution
859    /// (i.e. it is feasible and bounded), and the algorithm has found
860    /// optimal flow and node potentials (primal and dual solutions),
861    /// \n \c UNBOUNDED if the objective function of the problem is
862    /// unbounded, i.e. there is a directed cycle having negative total
863    /// cost and infinite upper bound.
864    ///
865    /// \see ProblemType, PivotRule
866    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
867      if (!init()) return INFEASIBLE;
868      return start(pivot_rule);
869    }
870
871    /// \brief Reset all the parameters that have been given before.
872    ///
873    /// This function resets all the paramaters that have been given
874    /// before using functions \ref lowerMap(), \ref upperMap(),
875    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
876    ///
877    /// It is useful for multiple run() calls. If this function is not
878    /// used, all the parameters given before are kept for the next
879    /// \ref run() call.
880    /// However the underlying digraph must not be modified after this
881    /// class have been constructed, since it copies and extends the graph.
882    ///
883    /// For example,
884    /// \code
885    ///   NetworkSimplex<ListDigraph> ns(graph);
886    ///
887    ///   // First run
888    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
889    ///     .supplyMap(sup).run();
890    ///
891    ///   // Run again with modified cost map (reset() is not called,
892    ///   // so only the cost map have to be set again)
893    ///   cost[e] += 100;
894    ///   ns.costMap(cost).run();
895    ///
896    ///   // Run again from scratch using reset()
897    ///   // (the lower bounds will be set to zero on all arcs)
898    ///   ns.reset();
899    ///   ns.upperMap(capacity).costMap(cost)
900    ///     .supplyMap(sup).run();
901    /// \endcode
902    ///
903    /// \return <tt>(*this)</tt>
904    NetworkSimplex& reset() {
905      for (int i = 0; i != _node_num; ++i) {
906        _supply[i] = 0;
907      }
908      for (int i = 0; i != _arc_num; ++i) {
909        _lower[i] = 0;
910        _upper[i] = INF;
911        _cost[i] = 1;
912      }
913      _have_lower = false;
914      _stype = GEQ;
915      return *this;
916    }
917
918    /// @}
919
920    /// \name Query Functions
921    /// The results of the algorithm can be obtained using these
922    /// functions.\n
923    /// The \ref run() function must be called before using them.
924
925    /// @{
926
927    /// \brief Return the total cost of the found flow.
928    ///
929    /// This function returns the total cost of the found flow.
930    /// Its complexity is O(e).
931    ///
932    /// \note The return type of the function can be specified as a
933    /// template parameter. For example,
934    /// \code
935    ///   ns.totalCost<double>();
936    /// \endcode
937    /// It is useful if the total cost cannot be stored in the \c Cost
938    /// type of the algorithm, which is the default return type of the
939    /// function.
940    ///
941    /// \pre \ref run() must be called before using this function.
942    template <typename Number>
943    Number totalCost() const {
944      Number c = 0;
945      for (ArcIt a(_graph); a != INVALID; ++a) {
946        int i = _arc_id[a];
947        c += Number(_flow[i]) * Number(_cost[i]);
948      }
949      return c;
950    }
951
952#ifndef DOXYGEN
953    Cost totalCost() const {
954      return totalCost<Cost>();
955    }
956#endif
957
958    /// \brief Return the flow on the given arc.
959    ///
960    /// This function returns the flow on the given arc.
961    ///
962    /// \pre \ref run() must be called before using this function.
963    Value flow(const Arc& a) const {
964      return _flow[_arc_id[a]];
965    }
966
967    /// \brief Return the flow map (the primal solution).
968    ///
969    /// This function copies the flow value on each arc into the given
970    /// map. The \c Value type of the algorithm must be convertible to
971    /// the \c Value type of the map.
972    ///
973    /// \pre \ref run() must be called before using this function.
974    template <typename FlowMap>
975    void flowMap(FlowMap &map) const {
976      for (ArcIt a(_graph); a != INVALID; ++a) {
977        map.set(a, _flow[_arc_id[a]]);
978      }
979    }
980
981    /// \brief Return the potential (dual value) of the given node.
982    ///
983    /// This function returns the potential (dual value) of the
984    /// given node.
985    ///
986    /// \pre \ref run() must be called before using this function.
987    Cost potential(const Node& n) const {
988      return _pi[_node_id[n]];
989    }
990
991    /// \brief Return the potential map (the dual solution).
992    ///
993    /// This function copies the potential (dual value) of each node
994    /// into the given map.
995    /// The \c Cost type of the algorithm must be convertible to the
996    /// \c Value type of the map.
997    ///
998    /// \pre \ref run() must be called before using this function.
999    template <typename PotentialMap>
1000    void potentialMap(PotentialMap &map) const {
1001      for (NodeIt n(_graph); n != INVALID; ++n) {
1002        map.set(n, _pi[_node_id[n]]);
1003      }
1004    }
1005
1006    /// @}
1007
1008  private:
1009
1010    // Initialize internal data structures
1011    bool init() {
1012      if (_node_num == 0) return false;
1013
1014      // Check the sum of supply values
1015      _sum_supply = 0;
1016      for (int i = 0; i != _node_num; ++i) {
1017        _sum_supply += _supply[i];
1018      }
1019      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1020             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1021
1022      // Remove non-zero lower bounds
1023      if (_have_lower) {
1024        for (int i = 0; i != _arc_num; ++i) {
1025          Value c = _lower[i];
1026          if (c >= 0) {
1027            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1028          } else {
1029            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1030          }
1031          _supply[_source[i]] -= c;
1032          _supply[_target[i]] += c;
1033        }
1034      } else {
1035        for (int i = 0; i != _arc_num; ++i) {
1036          _cap[i] = _upper[i];
1037        }
1038      }
1039
1040      // Initialize artifical cost
1041      Cost ART_COST;
1042      if (std::numeric_limits<Cost>::is_exact) {
1043        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1044      } else {
1045        ART_COST = std::numeric_limits<Cost>::min();
1046        for (int i = 0; i != _arc_num; ++i) {
1047          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1048        }
1049        ART_COST = (ART_COST + 1) * _node_num;
1050      }
1051
1052      // Initialize arc maps
1053      for (int i = 0; i != _arc_num; ++i) {
1054        _flow[i] = 0;
1055        _state[i] = STATE_LOWER;
1056      }
1057     
1058      // Set data for the artificial root node
1059      _root = _node_num;
1060      _parent[_root] = -1;
1061      _pred[_root] = -1;
1062      _thread[_root] = 0;
1063      _rev_thread[0] = _root;
1064      _succ_num[_root] = _node_num + 1;
1065      _last_succ[_root] = _root - 1;
1066      _supply[_root] = -_sum_supply;
1067      _pi[_root] = 0;
1068
1069      // Add artificial arcs and initialize the spanning tree data structure
1070      if (_sum_supply == 0) {
1071        // EQ supply constraints
1072        _search_arc_num = _arc_num;
1073        _all_arc_num = _arc_num + _node_num;
1074        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1075          _parent[u] = _root;
1076          _pred[u] = e;
1077          _thread[u] = u + 1;
1078          _rev_thread[u + 1] = u;
1079          _succ_num[u] = 1;
1080          _last_succ[u] = u;
1081          _cap[e] = INF;
1082          _state[e] = STATE_TREE;
1083          if (_supply[u] >= 0) {
1084            _forward[u] = true;
1085            _pi[u] = 0;
1086            _source[e] = u;
1087            _target[e] = _root;
1088            _flow[e] = _supply[u];
1089            _cost[e] = 0;
1090          } else {
1091            _forward[u] = false;
1092            _pi[u] = ART_COST;
1093            _source[e] = _root;
1094            _target[e] = u;
1095            _flow[e] = -_supply[u];
1096            _cost[e] = ART_COST;
1097          }
1098        }
1099      }
1100      else if (_sum_supply > 0) {
1101        // LEQ supply constraints
1102        _search_arc_num = _arc_num + _node_num;
1103        int f = _arc_num + _node_num;
1104        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1105          _parent[u] = _root;
1106          _thread[u] = u + 1;
1107          _rev_thread[u + 1] = u;
1108          _succ_num[u] = 1;
1109          _last_succ[u] = u;
1110          if (_supply[u] >= 0) {
1111            _forward[u] = true;
1112            _pi[u] = 0;
1113            _pred[u] = e;
1114            _source[e] = u;
1115            _target[e] = _root;
1116            _cap[e] = INF;
1117            _flow[e] = _supply[u];
1118            _cost[e] = 0;
1119            _state[e] = STATE_TREE;
1120          } else {
1121            _forward[u] = false;
1122            _pi[u] = ART_COST;
1123            _pred[u] = f;
1124            _source[f] = _root;
1125            _target[f] = u;
1126            _cap[f] = INF;
1127            _flow[f] = -_supply[u];
1128            _cost[f] = ART_COST;
1129            _state[f] = STATE_TREE;
1130            _source[e] = u;
1131            _target[e] = _root;
1132            _cap[e] = INF;
1133            _flow[e] = 0;
1134            _cost[e] = 0;
1135            _state[e] = STATE_LOWER;
1136            ++f;
1137          }
1138        }
1139        _all_arc_num = f;
1140      }
1141      else {
1142        // GEQ supply constraints
1143        _search_arc_num = _arc_num + _node_num;
1144        int f = _arc_num + _node_num;
1145        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1146          _parent[u] = _root;
1147          _thread[u] = u + 1;
1148          _rev_thread[u + 1] = u;
1149          _succ_num[u] = 1;
1150          _last_succ[u] = u;
1151          if (_supply[u] <= 0) {
1152            _forward[u] = false;
1153            _pi[u] = 0;
1154            _pred[u] = e;
1155            _source[e] = _root;
1156            _target[e] = u;
1157            _cap[e] = INF;
1158            _flow[e] = -_supply[u];
1159            _cost[e] = 0;
1160            _state[e] = STATE_TREE;
1161          } else {
1162            _forward[u] = true;
1163            _pi[u] = -ART_COST;
1164            _pred[u] = f;
1165            _source[f] = u;
1166            _target[f] = _root;
1167            _cap[f] = INF;
1168            _flow[f] = _supply[u];
1169            _state[f] = STATE_TREE;
1170            _cost[f] = ART_COST;
1171            _source[e] = _root;
1172            _target[e] = u;
1173            _cap[e] = INF;
1174            _flow[e] = 0;
1175            _cost[e] = 0;
1176            _state[e] = STATE_LOWER;
1177            ++f;
1178          }
1179        }
1180        _all_arc_num = f;
1181      }
1182
1183      return true;
1184    }
1185
1186    // Find the join node
1187    void findJoinNode() {
1188      int u = _source[in_arc];
1189      int v = _target[in_arc];
1190      while (u != v) {
1191        if (_succ_num[u] < _succ_num[v]) {
1192          u = _parent[u];
1193        } else {
1194          v = _parent[v];
1195        }
1196      }
1197      join = u;
1198    }
1199
1200    // Find the leaving arc of the cycle and returns true if the
1201    // leaving arc is not the same as the entering arc
1202    bool findLeavingArc() {
1203      // Initialize first and second nodes according to the direction
1204      // of the cycle
1205      if (_state[in_arc] == STATE_LOWER) {
1206        first  = _source[in_arc];
1207        second = _target[in_arc];
1208      } else {
1209        first  = _target[in_arc];
1210        second = _source[in_arc];
1211      }
1212      delta = _cap[in_arc];
1213      int result = 0;
1214      Value d;
1215      int e;
1216
1217      // Search the cycle along the path form the first node to the root
1218      for (int u = first; u != join; u = _parent[u]) {
1219        e = _pred[u];
1220        d = _forward[u] ?
1221          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1222        if (d < delta) {
1223          delta = d;
1224          u_out = u;
1225          result = 1;
1226        }
1227      }
1228      // Search the cycle along the path form the second node to the root
1229      for (int u = second; u != join; u = _parent[u]) {
1230        e = _pred[u];
1231        d = _forward[u] ?
1232          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1233        if (d <= delta) {
1234          delta = d;
1235          u_out = u;
1236          result = 2;
1237        }
1238      }
1239
1240      if (result == 1) {
1241        u_in = first;
1242        v_in = second;
1243      } else {
1244        u_in = second;
1245        v_in = first;
1246      }
1247      return result != 0;
1248    }
1249
1250    // Change _flow and _state vectors
1251    void changeFlow(bool change) {
1252      // Augment along the cycle
1253      if (delta > 0) {
1254        Value val = _state[in_arc] * delta;
1255        _flow[in_arc] += val;
1256        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1257          _flow[_pred[u]] += _forward[u] ? -val : val;
1258        }
1259        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1260          _flow[_pred[u]] += _forward[u] ? val : -val;
1261        }
1262      }
1263      // Update the state of the entering and leaving arcs
1264      if (change) {
1265        _state[in_arc] = STATE_TREE;
1266        _state[_pred[u_out]] =
1267          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1268      } else {
1269        _state[in_arc] = -_state[in_arc];
1270      }
1271    }
1272
1273    // Update the tree structure
1274    void updateTreeStructure() {
1275      int u, w;
1276      int old_rev_thread = _rev_thread[u_out];
1277      int old_succ_num = _succ_num[u_out];
1278      int old_last_succ = _last_succ[u_out];
1279      v_out = _parent[u_out];
1280
1281      u = _last_succ[u_in];  // the last successor of u_in
1282      right = _thread[u];    // the node after it
1283
1284      // Handle the case when old_rev_thread equals to v_in
1285      // (it also means that join and v_out coincide)
1286      if (old_rev_thread == v_in) {
1287        last = _thread[_last_succ[u_out]];
1288      } else {
1289        last = _thread[v_in];
1290      }
1291
1292      // Update _thread and _parent along the stem nodes (i.e. the nodes
1293      // between u_in and u_out, whose parent have to be changed)
1294      _thread[v_in] = stem = u_in;
1295      _dirty_revs.clear();
1296      _dirty_revs.push_back(v_in);
1297      par_stem = v_in;
1298      while (stem != u_out) {
1299        // Insert the next stem node into the thread list
1300        new_stem = _parent[stem];
1301        _thread[u] = new_stem;
1302        _dirty_revs.push_back(u);
1303
1304        // Remove the subtree of stem from the thread list
1305        w = _rev_thread[stem];
1306        _thread[w] = right;
1307        _rev_thread[right] = w;
1308
1309        // Change the parent node and shift stem nodes
1310        _parent[stem] = par_stem;
1311        par_stem = stem;
1312        stem = new_stem;
1313
1314        // Update u and right
1315        u = _last_succ[stem] == _last_succ[par_stem] ?
1316          _rev_thread[par_stem] : _last_succ[stem];
1317        right = _thread[u];
1318      }
1319      _parent[u_out] = par_stem;
1320      _thread[u] = last;
1321      _rev_thread[last] = u;
1322      _last_succ[u_out] = u;
1323
1324      // Remove the subtree of u_out from the thread list except for
1325      // the case when old_rev_thread equals to v_in
1326      // (it also means that join and v_out coincide)
1327      if (old_rev_thread != v_in) {
1328        _thread[old_rev_thread] = right;
1329        _rev_thread[right] = old_rev_thread;
1330      }
1331
1332      // Update _rev_thread using the new _thread values
1333      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1334        u = _dirty_revs[i];
1335        _rev_thread[_thread[u]] = u;
1336      }
1337
1338      // Update _pred, _forward, _last_succ and _succ_num for the
1339      // stem nodes from u_out to u_in
1340      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1341      u = u_out;
1342      while (u != u_in) {
1343        w = _parent[u];
1344        _pred[u] = _pred[w];
1345        _forward[u] = !_forward[w];
1346        tmp_sc += _succ_num[u] - _succ_num[w];
1347        _succ_num[u] = tmp_sc;
1348        _last_succ[w] = tmp_ls;
1349        u = w;
1350      }
1351      _pred[u_in] = in_arc;
1352      _forward[u_in] = (u_in == _source[in_arc]);
1353      _succ_num[u_in] = old_succ_num;
1354
1355      // Set limits for updating _last_succ form v_in and v_out
1356      // towards the root
1357      int up_limit_in = -1;
1358      int up_limit_out = -1;
1359      if (_last_succ[join] == v_in) {
1360        up_limit_out = join;
1361      } else {
1362        up_limit_in = join;
1363      }
1364
1365      // Update _last_succ from v_in towards the root
1366      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1367           u = _parent[u]) {
1368        _last_succ[u] = _last_succ[u_out];
1369      }
1370      // Update _last_succ from v_out towards the root
1371      if (join != old_rev_thread && v_in != old_rev_thread) {
1372        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1373             u = _parent[u]) {
1374          _last_succ[u] = old_rev_thread;
1375        }
1376      } else {
1377        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1378             u = _parent[u]) {
1379          _last_succ[u] = _last_succ[u_out];
1380        }
1381      }
1382
1383      // Update _succ_num from v_in to join
1384      for (u = v_in; u != join; u = _parent[u]) {
1385        _succ_num[u] += old_succ_num;
1386      }
1387      // Update _succ_num from v_out to join
1388      for (u = v_out; u != join; u = _parent[u]) {
1389        _succ_num[u] -= old_succ_num;
1390      }
1391    }
1392
1393    // Update potentials
1394    void updatePotential() {
1395      Cost sigma = _forward[u_in] ?
1396        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1397        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1398      // Update potentials in the subtree, which has been moved
1399      int end = _thread[_last_succ[u_in]];
1400      for (int u = u_in; u != end; u = _thread[u]) {
1401        _pi[u] += sigma;
1402      }
1403    }
1404
1405    // Execute the algorithm
1406    ProblemType start(PivotRule pivot_rule) {
1407      // Select the pivot rule implementation
1408      switch (pivot_rule) {
1409        case FIRST_ELIGIBLE:
1410          return start<FirstEligiblePivotRule>();
1411        case BEST_ELIGIBLE:
1412          return start<BestEligiblePivotRule>();
1413        case BLOCK_SEARCH:
1414          return start<BlockSearchPivotRule>();
1415        case CANDIDATE_LIST:
1416          return start<CandidateListPivotRule>();
1417        case ALTERING_LIST:
1418          return start<AlteringListPivotRule>();
1419      }
1420      return INFEASIBLE; // avoid warning
1421    }
1422
1423    template <typename PivotRuleImpl>
1424    ProblemType start() {
1425      PivotRuleImpl pivot(*this);
1426
1427      // Execute the Network Simplex algorithm
1428      while (pivot.findEnteringArc()) {
1429        findJoinNode();
1430        bool change = findLeavingArc();
1431        if (delta >= INF) return UNBOUNDED;
1432        changeFlow(change);
1433        if (change) {
1434          updateTreeStructure();
1435          updatePotential();
1436        }
1437      }
1438     
1439      // Check feasibility
1440      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1441        if (_flow[e] != 0) return INFEASIBLE;
1442      }
1443
1444      // Transform the solution and the supply map to the original form
1445      if (_have_lower) {
1446        for (int i = 0; i != _arc_num; ++i) {
1447          Value c = _lower[i];
1448          if (c != 0) {
1449            _flow[i] += c;
1450            _supply[_source[i]] += c;
1451            _supply[_target[i]] -= c;
1452          }
1453        }
1454      }
1455     
1456      // Shift potentials to meet the requirements of the GEQ/LEQ type
1457      // optimality conditions
1458      if (_sum_supply == 0) {
1459        if (_stype == GEQ) {
1460          Cost max_pot = std::numeric_limits<Cost>::min();
1461          for (int i = 0; i != _node_num; ++i) {
1462            if (_pi[i] > max_pot) max_pot = _pi[i];
1463          }
1464          if (max_pot > 0) {
1465            for (int i = 0; i != _node_num; ++i)
1466              _pi[i] -= max_pot;
1467          }
1468        } else {
1469          Cost min_pot = std::numeric_limits<Cost>::max();
1470          for (int i = 0; i != _node_num; ++i) {
1471            if (_pi[i] < min_pot) min_pot = _pi[i];
1472          }
1473          if (min_pot < 0) {
1474            for (int i = 0; i != _node_num; ++i)
1475              _pi[i] -= min_pot;
1476          }
1477        }
1478      }
1479
1480      return OPTIMAL;
1481    }
1482
1483  }; //class NetworkSimplex
1484
1485  ///@}
1486
1487} //namespace lemon
1488
1489#endif //LEMON_NETWORK_SIMPLEX_H
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