COIN-OR::LEMON - Graph Library

source: lemon/lemon/cycle_canceling.h @ 1165:16f55008c863

Last change on this file since 1165:16f55008c863 was 1165:16f55008c863, checked in by Peter Kovacs <kpeter@…>, 8 years ago

Doc improvements for min cost flow algorithms (#437)

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2010
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_CYCLE_CANCELING_H
20#define LEMON_CYCLE_CANCELING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
25
26#include <vector>
27#include <limits>
28
29#include <lemon/core.h>
30#include <lemon/maps.h>
31#include <lemon/path.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/adaptors.h>
35#include <lemon/circulation.h>
36#include <lemon/bellman_ford.h>
37#include <lemon/howard_mmc.h>
38
39namespace lemon {
40
41  /// \addtogroup min_cost_flow_algs
42  /// @{
43
44  /// \brief Implementation of cycle-canceling algorithms for
45  /// finding a \ref min_cost_flow "minimum cost flow".
46  ///
47  /// \ref CycleCanceling implements three different cycle-canceling
48  /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
49  /// \ref amo93networkflows, \ref klein67primal,
50  /// \ref goldberg89cyclecanceling.
51  /// The most efficent one is the \ref CANCEL_AND_TIGHTEN
52  /// "Cancel-and-Tighten" algorithm, thus it is the default method.
53  /// It runs in strongly polynomial time, but in practice, it is typically
54  /// orders of magnitude slower than the scaling algorithms and
55  /// \ref NetworkSimplex.
56  /// (For more information, see \ref min_cost_flow_algs "the module page".)
57  ///
58  /// Most of the parameters of the problem (except for the digraph)
59  /// can be given using separate functions, and the algorithm can be
60  /// executed using the \ref run() function. If some parameters are not
61  /// specified, then default values will be used.
62  ///
63  /// \tparam GR The digraph type the algorithm runs on.
64  /// \tparam V The number type used for flow amounts, capacity bounds
65  /// and supply values in the algorithm. By default, it is \c int.
66  /// \tparam C The number type used for costs and potentials in the
67  /// algorithm. By default, it is the same as \c V.
68  ///
69  /// \warning Both \c V and \c C must be signed number types.
70  /// \warning All input data (capacities, supply values, and costs) must
71  /// be integer.
72  /// \warning This algorithm does not support negative costs for
73  /// arcs having infinite upper bound.
74  ///
75  /// \note For more information about the three available methods,
76  /// see \ref Method.
77#ifdef DOXYGEN
78  template <typename GR, typename V, typename C>
79#else
80  template <typename GR, typename V = int, typename C = V>
81#endif
82  class CycleCanceling
83  {
84  public:
85
86    /// The type of the digraph
87    typedef GR Digraph;
88    /// The type of the flow amounts, capacity bounds and supply values
89    typedef V Value;
90    /// The type of the arc costs
91    typedef C Cost;
92
93  public:
94
95    /// \brief Problem type constants for the \c run() function.
96    ///
97    /// Enum type containing the problem type constants that can be
98    /// returned by the \ref run() function of the algorithm.
99    enum ProblemType {
100      /// The problem has no feasible solution (flow).
101      INFEASIBLE,
102      /// The problem has optimal solution (i.e. it is feasible and
103      /// bounded), and the algorithm has found optimal flow and node
104      /// potentials (primal and dual solutions).
105      OPTIMAL,
106      /// The digraph contains an arc of negative cost and infinite
107      /// upper bound. It means that the objective function is unbounded
108      /// on that arc, however, note that it could actually be bounded
109      /// over the feasible flows, but this algroithm cannot handle
110      /// these cases.
111      UNBOUNDED
112    };
113
114    /// \brief Constants for selecting the used method.
115    ///
116    /// Enum type containing constants for selecting the used method
117    /// for the \ref run() function.
118    ///
119    /// \ref CycleCanceling provides three different cycle-canceling
120    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten"
121    /// is used, which is by far the most efficient and the most robust.
122    /// However, the other methods can be selected using the \ref run()
123    /// function with the proper parameter.
124    enum Method {
125      /// A simple cycle-canceling method, which uses the
126      /// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative
127      /// cycles in the residual network.
128      /// The number of Bellman-Ford iterations is bounded by a successively
129      /// increased limit.
130      SIMPLE_CYCLE_CANCELING,
131      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
132      /// well-known strongly polynomial method
133      /// \ref goldberg89cyclecanceling. It improves along a
134      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
135      /// Its running time complexity is O(n<sup>2</sup>e<sup>3</sup>log(n)).
136      MINIMUM_MEAN_CYCLE_CANCELING,
137      /// The "Cancel-and-Tighten" algorithm, which can be viewed as an
138      /// improved version of the previous method
139      /// \ref goldberg89cyclecanceling.
140      /// It is faster both in theory and in practice, its running time
141      /// complexity is O(n<sup>2</sup>e<sup>2</sup>log(n)).
142      CANCEL_AND_TIGHTEN
143    };
144
145  private:
146
147    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
148
149    typedef std::vector<int> IntVector;
150    typedef std::vector<double> DoubleVector;
151    typedef std::vector<Value> ValueVector;
152    typedef std::vector<Cost> CostVector;
153    typedef std::vector<char> BoolVector;
154    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
155
156  private:
157
158    template <typename KT, typename VT>
159    class StaticVectorMap {
160    public:
161      typedef KT Key;
162      typedef VT Value;
163
164      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
165
166      const Value& operator[](const Key& key) const {
167        return _v[StaticDigraph::id(key)];
168      }
169
170      Value& operator[](const Key& key) {
171        return _v[StaticDigraph::id(key)];
172      }
173
174      void set(const Key& key, const Value& val) {
175        _v[StaticDigraph::id(key)] = val;
176      }
177
178    private:
179      std::vector<Value>& _v;
180    };
181
182    typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
183    typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
184
185  private:
186
187
188    // Data related to the underlying digraph
189    const GR &_graph;
190    int _node_num;
191    int _arc_num;
192    int _res_node_num;
193    int _res_arc_num;
194    int _root;
195
196    // Parameters of the problem
197    bool _have_lower;
198    Value _sum_supply;
199
200    // Data structures for storing the digraph
201    IntNodeMap _node_id;
202    IntArcMap _arc_idf;
203    IntArcMap _arc_idb;
204    IntVector _first_out;
205    BoolVector _forward;
206    IntVector _source;
207    IntVector _target;
208    IntVector _reverse;
209
210    // Node and arc data
211    ValueVector _lower;
212    ValueVector _upper;
213    CostVector _cost;
214    ValueVector _supply;
215
216    ValueVector _res_cap;
217    CostVector _pi;
218
219    // Data for a StaticDigraph structure
220    typedef std::pair<int, int> IntPair;
221    StaticDigraph _sgr;
222    std::vector<IntPair> _arc_vec;
223    std::vector<Cost> _cost_vec;
224    IntVector _id_vec;
225    CostArcMap _cost_map;
226    CostNodeMap _pi_map;
227
228  public:
229
230    /// \brief Constant for infinite upper bounds (capacities).
231    ///
232    /// Constant for infinite upper bounds (capacities).
233    /// It is \c std::numeric_limits<Value>::infinity() if available,
234    /// \c std::numeric_limits<Value>::max() otherwise.
235    const Value INF;
236
237  public:
238
239    /// \brief Constructor.
240    ///
241    /// The constructor of the class.
242    ///
243    /// \param graph The digraph the algorithm runs on.
244    CycleCanceling(const GR& graph) :
245      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
246      _cost_map(_cost_vec), _pi_map(_pi),
247      INF(std::numeric_limits<Value>::has_infinity ?
248          std::numeric_limits<Value>::infinity() :
249          std::numeric_limits<Value>::max())
250    {
251      // Check the number types
252      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
253        "The flow type of CycleCanceling must be signed");
254      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
255        "The cost type of CycleCanceling must be signed");
256
257      // Reset data structures
258      reset();
259    }
260
261    /// \name Parameters
262    /// The parameters of the algorithm can be specified using these
263    /// functions.
264
265    /// @{
266
267    /// \brief Set the lower bounds on the arcs.
268    ///
269    /// This function sets the lower bounds on the arcs.
270    /// If it is not used before calling \ref run(), the lower bounds
271    /// will be set to zero on all arcs.
272    ///
273    /// \param map An arc map storing the lower bounds.
274    /// Its \c Value type must be convertible to the \c Value type
275    /// of the algorithm.
276    ///
277    /// \return <tt>(*this)</tt>
278    template <typename LowerMap>
279    CycleCanceling& lowerMap(const LowerMap& map) {
280      _have_lower = true;
281      for (ArcIt a(_graph); a != INVALID; ++a) {
282        _lower[_arc_idf[a]] = map[a];
283        _lower[_arc_idb[a]] = map[a];
284      }
285      return *this;
286    }
287
288    /// \brief Set the upper bounds (capacities) on the arcs.
289    ///
290    /// This function sets the upper bounds (capacities) on the arcs.
291    /// If it is not used before calling \ref run(), the upper bounds
292    /// will be set to \ref INF on all arcs (i.e. the flow value will be
293    /// unbounded from above).
294    ///
295    /// \param map An arc map storing the upper bounds.
296    /// Its \c Value type must be convertible to the \c Value type
297    /// of the algorithm.
298    ///
299    /// \return <tt>(*this)</tt>
300    template<typename UpperMap>
301    CycleCanceling& upperMap(const UpperMap& map) {
302      for (ArcIt a(_graph); a != INVALID; ++a) {
303        _upper[_arc_idf[a]] = map[a];
304      }
305      return *this;
306    }
307
308    /// \brief Set the costs of the arcs.
309    ///
310    /// This function sets the costs of the arcs.
311    /// If it is not used before calling \ref run(), the costs
312    /// will be set to \c 1 on all arcs.
313    ///
314    /// \param map An arc map storing the costs.
315    /// Its \c Value type must be convertible to the \c Cost type
316    /// of the algorithm.
317    ///
318    /// \return <tt>(*this)</tt>
319    template<typename CostMap>
320    CycleCanceling& costMap(const CostMap& map) {
321      for (ArcIt a(_graph); a != INVALID; ++a) {
322        _cost[_arc_idf[a]] =  map[a];
323        _cost[_arc_idb[a]] = -map[a];
324      }
325      return *this;
326    }
327
328    /// \brief Set the supply values of the nodes.
329    ///
330    /// This function sets the supply values of the nodes.
331    /// If neither this function nor \ref stSupply() is used before
332    /// calling \ref run(), the supply of each node will be set to zero.
333    ///
334    /// \param map A node map storing the supply values.
335    /// Its \c Value type must be convertible to the \c Value type
336    /// of the algorithm.
337    ///
338    /// \return <tt>(*this)</tt>
339    template<typename SupplyMap>
340    CycleCanceling& supplyMap(const SupplyMap& map) {
341      for (NodeIt n(_graph); n != INVALID; ++n) {
342        _supply[_node_id[n]] = map[n];
343      }
344      return *this;
345    }
346
347    /// \brief Set single source and target nodes and a supply value.
348    ///
349    /// This function sets a single source node and a single target node
350    /// and the required flow value.
351    /// If neither this function nor \ref supplyMap() is used before
352    /// calling \ref run(), the supply of each node will be set to zero.
353    ///
354    /// Using this function has the same effect as using \ref supplyMap()
355    /// with a map in which \c k is assigned to \c s, \c -k is
356    /// assigned to \c t and all other nodes have zero supply value.
357    ///
358    /// \param s The source node.
359    /// \param t The target node.
360    /// \param k The required amount of flow from node \c s to node \c t
361    /// (i.e. the supply of \c s and the demand of \c t).
362    ///
363    /// \return <tt>(*this)</tt>
364    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
365      for (int i = 0; i != _res_node_num; ++i) {
366        _supply[i] = 0;
367      }
368      _supply[_node_id[s]] =  k;
369      _supply[_node_id[t]] = -k;
370      return *this;
371    }
372
373    /// @}
374
375    /// \name Execution control
376    /// The algorithm can be executed using \ref run().
377
378    /// @{
379
380    /// \brief Run the algorithm.
381    ///
382    /// This function runs the algorithm.
383    /// The paramters can be specified using functions \ref lowerMap(),
384    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
385    /// For example,
386    /// \code
387    ///   CycleCanceling<ListDigraph> cc(graph);
388    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
389    ///     .supplyMap(sup).run();
390    /// \endcode
391    ///
392    /// This function can be called more than once. All the given parameters
393    /// are kept for the next call, unless \ref resetParams() or \ref reset()
394    /// is used, thus only the modified parameters have to be set again.
395    /// If the underlying digraph was also modified after the construction
396    /// of the class (or the last \ref reset() call), then the \ref reset()
397    /// function must be called.
398    ///
399    /// \param method The cycle-canceling method that will be used.
400    /// For more information, see \ref Method.
401    ///
402    /// \return \c INFEASIBLE if no feasible flow exists,
403    /// \n \c OPTIMAL if the problem has optimal solution
404    /// (i.e. it is feasible and bounded), and the algorithm has found
405    /// optimal flow and node potentials (primal and dual solutions),
406    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
407    /// and infinite upper bound. It means that the objective function
408    /// is unbounded on that arc, however, note that it could actually be
409    /// bounded over the feasible flows, but this algroithm cannot handle
410    /// these cases.
411    ///
412    /// \see ProblemType, Method
413    /// \see resetParams(), reset()
414    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
415      ProblemType pt = init();
416      if (pt != OPTIMAL) return pt;
417      start(method);
418      return OPTIMAL;
419    }
420
421    /// \brief Reset all the parameters that have been given before.
422    ///
423    /// This function resets all the paramaters that have been given
424    /// before using functions \ref lowerMap(), \ref upperMap(),
425    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
426    ///
427    /// It is useful for multiple \ref run() calls. Basically, all the given
428    /// parameters are kept for the next \ref run() call, unless
429    /// \ref resetParams() or \ref reset() is used.
430    /// If the underlying digraph was also modified after the construction
431    /// of the class or the last \ref reset() call, then the \ref reset()
432    /// function must be used, otherwise \ref resetParams() is sufficient.
433    ///
434    /// For example,
435    /// \code
436    ///   CycleCanceling<ListDigraph> cs(graph);
437    ///
438    ///   // First run
439    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
440    ///     .supplyMap(sup).run();
441    ///
442    ///   // Run again with modified cost map (resetParams() is not called,
443    ///   // so only the cost map have to be set again)
444    ///   cost[e] += 100;
445    ///   cc.costMap(cost).run();
446    ///
447    ///   // Run again from scratch using resetParams()
448    ///   // (the lower bounds will be set to zero on all arcs)
449    ///   cc.resetParams();
450    ///   cc.upperMap(capacity).costMap(cost)
451    ///     .supplyMap(sup).run();
452    /// \endcode
453    ///
454    /// \return <tt>(*this)</tt>
455    ///
456    /// \see reset(), run()
457    CycleCanceling& resetParams() {
458      for (int i = 0; i != _res_node_num; ++i) {
459        _supply[i] = 0;
460      }
461      int limit = _first_out[_root];
462      for (int j = 0; j != limit; ++j) {
463        _lower[j] = 0;
464        _upper[j] = INF;
465        _cost[j] = _forward[j] ? 1 : -1;
466      }
467      for (int j = limit; j != _res_arc_num; ++j) {
468        _lower[j] = 0;
469        _upper[j] = INF;
470        _cost[j] = 0;
471        _cost[_reverse[j]] = 0;
472      }
473      _have_lower = false;
474      return *this;
475    }
476
477    /// \brief Reset the internal data structures and all the parameters
478    /// that have been given before.
479    ///
480    /// This function resets the internal data structures and all the
481    /// paramaters that have been given before using functions \ref lowerMap(),
482    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
483    ///
484    /// It is useful for multiple \ref run() calls. Basically, all the given
485    /// parameters are kept for the next \ref run() call, unless
486    /// \ref resetParams() or \ref reset() is used.
487    /// If the underlying digraph was also modified after the construction
488    /// of the class or the last \ref reset() call, then the \ref reset()
489    /// function must be used, otherwise \ref resetParams() is sufficient.
490    ///
491    /// See \ref resetParams() for examples.
492    ///
493    /// \return <tt>(*this)</tt>
494    ///
495    /// \see resetParams(), run()
496    CycleCanceling& reset() {
497      // Resize vectors
498      _node_num = countNodes(_graph);
499      _arc_num = countArcs(_graph);
500      _res_node_num = _node_num + 1;
501      _res_arc_num = 2 * (_arc_num + _node_num);
502      _root = _node_num;
503
504      _first_out.resize(_res_node_num + 1);
505      _forward.resize(_res_arc_num);
506      _source.resize(_res_arc_num);
507      _target.resize(_res_arc_num);
508      _reverse.resize(_res_arc_num);
509
510      _lower.resize(_res_arc_num);
511      _upper.resize(_res_arc_num);
512      _cost.resize(_res_arc_num);
513      _supply.resize(_res_node_num);
514
515      _res_cap.resize(_res_arc_num);
516      _pi.resize(_res_node_num);
517
518      _arc_vec.reserve(_res_arc_num);
519      _cost_vec.reserve(_res_arc_num);
520      _id_vec.reserve(_res_arc_num);
521
522      // Copy the graph
523      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
524      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
525        _node_id[n] = i;
526      }
527      i = 0;
528      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
529        _first_out[i] = j;
530        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
531          _arc_idf[a] = j;
532          _forward[j] = true;
533          _source[j] = i;
534          _target[j] = _node_id[_graph.runningNode(a)];
535        }
536        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
537          _arc_idb[a] = j;
538          _forward[j] = false;
539          _source[j] = i;
540          _target[j] = _node_id[_graph.runningNode(a)];
541        }
542        _forward[j] = false;
543        _source[j] = i;
544        _target[j] = _root;
545        _reverse[j] = k;
546        _forward[k] = true;
547        _source[k] = _root;
548        _target[k] = i;
549        _reverse[k] = j;
550        ++j; ++k;
551      }
552      _first_out[i] = j;
553      _first_out[_res_node_num] = k;
554      for (ArcIt a(_graph); a != INVALID; ++a) {
555        int fi = _arc_idf[a];
556        int bi = _arc_idb[a];
557        _reverse[fi] = bi;
558        _reverse[bi] = fi;
559      }
560
561      // Reset parameters
562      resetParams();
563      return *this;
564    }
565
566    /// @}
567
568    /// \name Query Functions
569    /// The results of the algorithm can be obtained using these
570    /// functions.\n
571    /// The \ref run() function must be called before using them.
572
573    /// @{
574
575    /// \brief Return the total cost of the found flow.
576    ///
577    /// This function returns the total cost of the found flow.
578    /// Its complexity is O(e).
579    ///
580    /// \note The return type of the function can be specified as a
581    /// template parameter. For example,
582    /// \code
583    ///   cc.totalCost<double>();
584    /// \endcode
585    /// It is useful if the total cost cannot be stored in the \c Cost
586    /// type of the algorithm, which is the default return type of the
587    /// function.
588    ///
589    /// \pre \ref run() must be called before using this function.
590    template <typename Number>
591    Number totalCost() const {
592      Number c = 0;
593      for (ArcIt a(_graph); a != INVALID; ++a) {
594        int i = _arc_idb[a];
595        c += static_cast<Number>(_res_cap[i]) *
596             (-static_cast<Number>(_cost[i]));
597      }
598      return c;
599    }
600
601#ifndef DOXYGEN
602    Cost totalCost() const {
603      return totalCost<Cost>();
604    }
605#endif
606
607    /// \brief Return the flow on the given arc.
608    ///
609    /// This function returns the flow on the given arc.
610    ///
611    /// \pre \ref run() must be called before using this function.
612    Value flow(const Arc& a) const {
613      return _res_cap[_arc_idb[a]];
614    }
615
616    /// \brief Copy the flow values (the primal solution) into the
617    /// given map.
618    ///
619    /// This function copies the flow value on each arc into the given
620    /// map. The \c Value type of the algorithm must be convertible to
621    /// the \c Value type of the map.
622    ///
623    /// \pre \ref run() must be called before using this function.
624    template <typename FlowMap>
625    void flowMap(FlowMap &map) const {
626      for (ArcIt a(_graph); a != INVALID; ++a) {
627        map.set(a, _res_cap[_arc_idb[a]]);
628      }
629    }
630
631    /// \brief Return the potential (dual value) of the given node.
632    ///
633    /// This function returns the potential (dual value) of the
634    /// given node.
635    ///
636    /// \pre \ref run() must be called before using this function.
637    Cost potential(const Node& n) const {
638      return static_cast<Cost>(_pi[_node_id[n]]);
639    }
640
641    /// \brief Copy the potential values (the dual solution) into the
642    /// given map.
643    ///
644    /// This function copies the potential (dual value) of each node
645    /// into the given map.
646    /// The \c Cost type of the algorithm must be convertible to the
647    /// \c Value type of the map.
648    ///
649    /// \pre \ref run() must be called before using this function.
650    template <typename PotentialMap>
651    void potentialMap(PotentialMap &map) const {
652      for (NodeIt n(_graph); n != INVALID; ++n) {
653        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
654      }
655    }
656
657    /// @}
658
659  private:
660
661    // Initialize the algorithm
662    ProblemType init() {
663      if (_res_node_num <= 1) return INFEASIBLE;
664
665      // Check the sum of supply values
666      _sum_supply = 0;
667      for (int i = 0; i != _root; ++i) {
668        _sum_supply += _supply[i];
669      }
670      if (_sum_supply > 0) return INFEASIBLE;
671
672
673      // Initialize vectors
674      for (int i = 0; i != _res_node_num; ++i) {
675        _pi[i] = 0;
676      }
677      ValueVector excess(_supply);
678
679      // Remove infinite upper bounds and check negative arcs
680      const Value MAX = std::numeric_limits<Value>::max();
681      int last_out;
682      if (_have_lower) {
683        for (int i = 0; i != _root; ++i) {
684          last_out = _first_out[i+1];
685          for (int j = _first_out[i]; j != last_out; ++j) {
686            if (_forward[j]) {
687              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
688              if (c >= MAX) return UNBOUNDED;
689              excess[i] -= c;
690              excess[_target[j]] += c;
691            }
692          }
693        }
694      } else {
695        for (int i = 0; i != _root; ++i) {
696          last_out = _first_out[i+1];
697          for (int j = _first_out[i]; j != last_out; ++j) {
698            if (_forward[j] && _cost[j] < 0) {
699              Value c = _upper[j];
700              if (c >= MAX) return UNBOUNDED;
701              excess[i] -= c;
702              excess[_target[j]] += c;
703            }
704          }
705        }
706      }
707      Value ex, max_cap = 0;
708      for (int i = 0; i != _res_node_num; ++i) {
709        ex = excess[i];
710        if (ex < 0) max_cap -= ex;
711      }
712      for (int j = 0; j != _res_arc_num; ++j) {
713        if (_upper[j] >= MAX) _upper[j] = max_cap;
714      }
715
716      // Initialize maps for Circulation and remove non-zero lower bounds
717      ConstMap<Arc, Value> low(0);
718      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
719      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
720      ValueArcMap cap(_graph), flow(_graph);
721      ValueNodeMap sup(_graph);
722      for (NodeIt n(_graph); n != INVALID; ++n) {
723        sup[n] = _supply[_node_id[n]];
724      }
725      if (_have_lower) {
726        for (ArcIt a(_graph); a != INVALID; ++a) {
727          int j = _arc_idf[a];
728          Value c = _lower[j];
729          cap[a] = _upper[j] - c;
730          sup[_graph.source(a)] -= c;
731          sup[_graph.target(a)] += c;
732        }
733      } else {
734        for (ArcIt a(_graph); a != INVALID; ++a) {
735          cap[a] = _upper[_arc_idf[a]];
736        }
737      }
738
739      // Find a feasible flow using Circulation
740      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
741        circ(_graph, low, cap, sup);
742      if (!circ.flowMap(flow).run()) return INFEASIBLE;
743
744      // Set residual capacities and handle GEQ supply type
745      if (_sum_supply < 0) {
746        for (ArcIt a(_graph); a != INVALID; ++a) {
747          Value fa = flow[a];
748          _res_cap[_arc_idf[a]] = cap[a] - fa;
749          _res_cap[_arc_idb[a]] = fa;
750          sup[_graph.source(a)] -= fa;
751          sup[_graph.target(a)] += fa;
752        }
753        for (NodeIt n(_graph); n != INVALID; ++n) {
754          excess[_node_id[n]] = sup[n];
755        }
756        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
757          int u = _target[a];
758          int ra = _reverse[a];
759          _res_cap[a] = -_sum_supply + 1;
760          _res_cap[ra] = -excess[u];
761          _cost[a] = 0;
762          _cost[ra] = 0;
763        }
764      } else {
765        for (ArcIt a(_graph); a != INVALID; ++a) {
766          Value fa = flow[a];
767          _res_cap[_arc_idf[a]] = cap[a] - fa;
768          _res_cap[_arc_idb[a]] = fa;
769        }
770        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
771          int ra = _reverse[a];
772          _res_cap[a] = 1;
773          _res_cap[ra] = 0;
774          _cost[a] = 0;
775          _cost[ra] = 0;
776        }
777      }
778
779      return OPTIMAL;
780    }
781
782    // Build a StaticDigraph structure containing the current
783    // residual network
784    void buildResidualNetwork() {
785      _arc_vec.clear();
786      _cost_vec.clear();
787      _id_vec.clear();
788      for (int j = 0; j != _res_arc_num; ++j) {
789        if (_res_cap[j] > 0) {
790          _arc_vec.push_back(IntPair(_source[j], _target[j]));
791          _cost_vec.push_back(_cost[j]);
792          _id_vec.push_back(j);
793        }
794      }
795      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
796    }
797
798    // Execute the algorithm and transform the results
799    void start(Method method) {
800      // Execute the algorithm
801      switch (method) {
802        case SIMPLE_CYCLE_CANCELING:
803          startSimpleCycleCanceling();
804          break;
805        case MINIMUM_MEAN_CYCLE_CANCELING:
806          startMinMeanCycleCanceling();
807          break;
808        case CANCEL_AND_TIGHTEN:
809          startCancelAndTighten();
810          break;
811      }
812
813      // Compute node potentials
814      if (method != SIMPLE_CYCLE_CANCELING) {
815        buildResidualNetwork();
816        typename BellmanFord<StaticDigraph, CostArcMap>
817          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
818        bf.distMap(_pi_map);
819        bf.init(0);
820        bf.start();
821      }
822
823      // Handle non-zero lower bounds
824      if (_have_lower) {
825        int limit = _first_out[_root];
826        for (int j = 0; j != limit; ++j) {
827          if (!_forward[j]) _res_cap[j] += _lower[j];
828        }
829      }
830    }
831
832    // Execute the "Simple Cycle Canceling" method
833    void startSimpleCycleCanceling() {
834      // Constants for computing the iteration limits
835      const int BF_FIRST_LIMIT  = 2;
836      const double BF_LIMIT_FACTOR = 1.5;
837
838      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
839      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
840      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
841      typedef typename BellmanFord<ResDigraph, CostArcMap>
842        ::template SetDistMap<CostNodeMap>
843        ::template SetPredMap<PredMap>::Create BF;
844
845      // Build the residual network
846      _arc_vec.clear();
847      _cost_vec.clear();
848      for (int j = 0; j != _res_arc_num; ++j) {
849        _arc_vec.push_back(IntPair(_source[j], _target[j]));
850        _cost_vec.push_back(_cost[j]);
851      }
852      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
853
854      FilterMap filter_map(_res_cap);
855      ResDigraph rgr(_sgr, filter_map);
856      std::vector<int> cycle;
857      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
858      PredMap pred_map(pred);
859      BF bf(rgr, _cost_map);
860      bf.distMap(_pi_map).predMap(pred_map);
861
862      int length_bound = BF_FIRST_LIMIT;
863      bool optimal = false;
864      while (!optimal) {
865        bf.init(0);
866        int iter_num = 0;
867        bool cycle_found = false;
868        while (!cycle_found) {
869          // Perform some iterations of the Bellman-Ford algorithm
870          int curr_iter_num = iter_num + length_bound <= _node_num ?
871            length_bound : _node_num - iter_num;
872          iter_num += curr_iter_num;
873          int real_iter_num = curr_iter_num;
874          for (int i = 0; i < curr_iter_num; ++i) {
875            if (bf.processNextWeakRound()) {
876              real_iter_num = i;
877              break;
878            }
879          }
880          if (real_iter_num < curr_iter_num) {
881            // Optimal flow is found
882            optimal = true;
883            break;
884          } else {
885            // Search for node disjoint negative cycles
886            std::vector<int> state(_res_node_num, 0);
887            int id = 0;
888            for (int u = 0; u != _res_node_num; ++u) {
889              if (state[u] != 0) continue;
890              ++id;
891              int v = u;
892              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
893                   -1 : rgr.id(rgr.source(pred[v]))) {
894                state[v] = id;
895              }
896              if (v != -1 && state[v] == id) {
897                // A negative cycle is found
898                cycle_found = true;
899                cycle.clear();
900                StaticDigraph::Arc a = pred[v];
901                Value d, delta = _res_cap[rgr.id(a)];
902                cycle.push_back(rgr.id(a));
903                while (rgr.id(rgr.source(a)) != v) {
904                  a = pred_map[rgr.source(a)];
905                  d = _res_cap[rgr.id(a)];
906                  if (d < delta) delta = d;
907                  cycle.push_back(rgr.id(a));
908                }
909
910                // Augment along the cycle
911                for (int i = 0; i < int(cycle.size()); ++i) {
912                  int j = cycle[i];
913                  _res_cap[j] -= delta;
914                  _res_cap[_reverse[j]] += delta;
915                }
916              }
917            }
918          }
919
920          // Increase iteration limit if no cycle is found
921          if (!cycle_found) {
922            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
923          }
924        }
925      }
926    }
927
928    // Execute the "Minimum Mean Cycle Canceling" method
929    void startMinMeanCycleCanceling() {
930      typedef SimplePath<StaticDigraph> SPath;
931      typedef typename SPath::ArcIt SPathArcIt;
932      typedef typename HowardMmc<StaticDigraph, CostArcMap>
933        ::template SetPath<SPath>::Create MMC;
934
935      SPath cycle;
936      MMC mmc(_sgr, _cost_map);
937      mmc.cycle(cycle);
938      buildResidualNetwork();
939      while (mmc.findCycleMean() && mmc.cycleCost() < 0) {
940        // Find the cycle
941        mmc.findCycle();
942
943        // Compute delta value
944        Value delta = INF;
945        for (SPathArcIt a(cycle); a != INVALID; ++a) {
946          Value d = _res_cap[_id_vec[_sgr.id(a)]];
947          if (d < delta) delta = d;
948        }
949
950        // Augment along the cycle
951        for (SPathArcIt a(cycle); a != INVALID; ++a) {
952          int j = _id_vec[_sgr.id(a)];
953          _res_cap[j] -= delta;
954          _res_cap[_reverse[j]] += delta;
955        }
956
957        // Rebuild the residual network
958        buildResidualNetwork();
959      }
960    }
961
962    // Execute the "Cancel-and-Tighten" method
963    void startCancelAndTighten() {
964      // Constants for the min mean cycle computations
965      const double LIMIT_FACTOR = 1.0;
966      const int MIN_LIMIT = 5;
967
968      // Contruct auxiliary data vectors
969      DoubleVector pi(_res_node_num, 0.0);
970      IntVector level(_res_node_num);
971      BoolVector reached(_res_node_num);
972      BoolVector processed(_res_node_num);
973      IntVector pred_node(_res_node_num);
974      IntVector pred_arc(_res_node_num);
975      std::vector<int> stack(_res_node_num);
976      std::vector<int> proc_vector(_res_node_num);
977
978      // Initialize epsilon
979      double epsilon = 0;
980      for (int a = 0; a != _res_arc_num; ++a) {
981        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
982          epsilon = -_cost[a];
983      }
984
985      // Start phases
986      Tolerance<double> tol;
987      tol.epsilon(1e-6);
988      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
989      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
990      int iter = limit;
991      while (epsilon * _res_node_num >= 1) {
992        // Find and cancel cycles in the admissible network using DFS
993        for (int u = 0; u != _res_node_num; ++u) {
994          reached[u] = false;
995          processed[u] = false;
996        }
997        int stack_head = -1;
998        int proc_head = -1;
999        for (int start = 0; start != _res_node_num; ++start) {
1000          if (reached[start]) continue;
1001
1002          // New start node
1003          reached[start] = true;
1004          pred_arc[start] = -1;
1005          pred_node[start] = -1;
1006
1007          // Find the first admissible outgoing arc
1008          double p = pi[start];
1009          int a = _first_out[start];
1010          int last_out = _first_out[start+1];
1011          for (; a != last_out && (_res_cap[a] == 0 ||
1012               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1013          if (a == last_out) {
1014            processed[start] = true;
1015            proc_vector[++proc_head] = start;
1016            continue;
1017          }
1018          stack[++stack_head] = a;
1019
1020          while (stack_head >= 0) {
1021            int sa = stack[stack_head];
1022            int u = _source[sa];
1023            int v = _target[sa];
1024
1025            if (!reached[v]) {
1026              // A new node is reached
1027              reached[v] = true;
1028              pred_node[v] = u;
1029              pred_arc[v] = sa;
1030              p = pi[v];
1031              a = _first_out[v];
1032              last_out = _first_out[v+1];
1033              for (; a != last_out && (_res_cap[a] == 0 ||
1034                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1035              stack[++stack_head] = a == last_out ? -1 : a;
1036            } else {
1037              if (!processed[v]) {
1038                // A cycle is found
1039                int n, w = u;
1040                Value d, delta = _res_cap[sa];
1041                for (n = u; n != v; n = pred_node[n]) {
1042                  d = _res_cap[pred_arc[n]];
1043                  if (d <= delta) {
1044                    delta = d;
1045                    w = pred_node[n];
1046                  }
1047                }
1048
1049                // Augment along the cycle
1050                _res_cap[sa] -= delta;
1051                _res_cap[_reverse[sa]] += delta;
1052                for (n = u; n != v; n = pred_node[n]) {
1053                  int pa = pred_arc[n];
1054                  _res_cap[pa] -= delta;
1055                  _res_cap[_reverse[pa]] += delta;
1056                }
1057                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1058                  --stack_head;
1059                  reached[n] = false;
1060                }
1061                u = w;
1062              }
1063              v = u;
1064
1065              // Find the next admissible outgoing arc
1066              p = pi[v];
1067              a = stack[stack_head] + 1;
1068              last_out = _first_out[v+1];
1069              for (; a != last_out && (_res_cap[a] == 0 ||
1070                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1071              stack[stack_head] = a == last_out ? -1 : a;
1072            }
1073
1074            while (stack_head >= 0 && stack[stack_head] == -1) {
1075              processed[v] = true;
1076              proc_vector[++proc_head] = v;
1077              if (--stack_head >= 0) {
1078                // Find the next admissible outgoing arc
1079                v = _source[stack[stack_head]];
1080                p = pi[v];
1081                a = stack[stack_head] + 1;
1082                last_out = _first_out[v+1];
1083                for (; a != last_out && (_res_cap[a] == 0 ||
1084                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1085                stack[stack_head] = a == last_out ? -1 : a;
1086              }
1087            }
1088          }
1089        }
1090
1091        // Tighten potentials and epsilon
1092        if (--iter > 0) {
1093          for (int u = 0; u != _res_node_num; ++u) {
1094            level[u] = 0;
1095          }
1096          for (int i = proc_head; i > 0; --i) {
1097            int u = proc_vector[i];
1098            double p = pi[u];
1099            int l = level[u] + 1;
1100            int last_out = _first_out[u+1];
1101            for (int a = _first_out[u]; a != last_out; ++a) {
1102              int v = _target[a];
1103              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1104                  l > level[v]) level[v] = l;
1105            }
1106          }
1107
1108          // Modify potentials
1109          double q = std::numeric_limits<double>::max();
1110          for (int u = 0; u != _res_node_num; ++u) {
1111            int lu = level[u];
1112            double p, pu = pi[u];
1113            int last_out = _first_out[u+1];
1114            for (int a = _first_out[u]; a != last_out; ++a) {
1115              if (_res_cap[a] == 0) continue;
1116              int v = _target[a];
1117              int ld = lu - level[v];
1118              if (ld > 0) {
1119                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1120                if (p < q) q = p;
1121              }
1122            }
1123          }
1124          for (int u = 0; u != _res_node_num; ++u) {
1125            pi[u] -= q * level[u];
1126          }
1127
1128          // Modify epsilon
1129          epsilon = 0;
1130          for (int u = 0; u != _res_node_num; ++u) {
1131            double curr, pu = pi[u];
1132            int last_out = _first_out[u+1];
1133            for (int a = _first_out[u]; a != last_out; ++a) {
1134              if (_res_cap[a] == 0) continue;
1135              curr = _cost[a] + pu - pi[_target[a]];
1136              if (-curr > epsilon) epsilon = -curr;
1137            }
1138          }
1139        } else {
1140          typedef HowardMmc<StaticDigraph, CostArcMap> MMC;
1141          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1142            ::template SetDistMap<CostNodeMap>::Create BF;
1143
1144          // Set epsilon to the minimum cycle mean
1145          buildResidualNetwork();
1146          MMC mmc(_sgr, _cost_map);
1147          mmc.findCycleMean();
1148          epsilon = -mmc.cycleMean();
1149          Cost cycle_cost = mmc.cycleCost();
1150          int cycle_size = mmc.cycleSize();
1151
1152          // Compute feasible potentials for the current epsilon
1153          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1154            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1155          }
1156          BF bf(_sgr, _cost_map);
1157          bf.distMap(_pi_map);
1158          bf.init(0);
1159          bf.start();
1160          for (int u = 0; u != _res_node_num; ++u) {
1161            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1162          }
1163
1164          iter = limit;
1165        }
1166      }
1167    }
1168
1169  }; //class CycleCanceling
1170
1171  ///@}
1172
1173} //namespace lemon
1174
1175#endif //LEMON_CYCLE_CANCELING_H
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