Add fractional matching algorithms (#314)
authorBalazs Dezso <deba@inf.elte.hu>
Fri, 25 Sep 2009 21:51:36 +0200
changeset 869636dadefe1e6
parent 868 0513ccfea967
child 870 61120524af27
Add fractional matching algorithms (#314)
doc/groups.dox
lemon/Makefile.am
lemon/fractional_matching.h
test/CMakeLists.txt
test/Makefile.am
test/fractional_matching_test.cc
     1.1 --- a/doc/groups.dox	Sun Sep 20 21:38:24 2009 +0200
     1.2 +++ b/doc/groups.dox	Fri Sep 25 21:51:36 2009 +0200
     1.3 @@ -349,7 +349,7 @@
     1.4  also provide functions to query the minimum cut, which is the dual
     1.5  problem of maximum flow.
     1.6  
     1.7 -\ref Circulation is a preflow push-relabel algorithm implemented directly 
     1.8 +\ref Circulation is a preflow push-relabel algorithm implemented directly
     1.9  for finding feasible circulations, which is a somewhat different problem,
    1.10  but it is strongly related to maximum flow.
    1.11  For more information, see \ref Circulation.
    1.12 @@ -470,6 +470,13 @@
    1.13  - \ref MaxWeightedPerfectMatching
    1.14    Edmond's blossom shrinking algorithm for calculating maximum weighted
    1.15    perfect matching in general graphs.
    1.16 +- \ref MaxFractionalMatching Push-relabel algorithm for calculating
    1.17 +  maximum cardinality fractional matching in general graphs.
    1.18 +- \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating
    1.19 +  maximum weighted fractional matching in general graphs.
    1.20 +- \ref MaxWeightedPerfectFractionalMatching
    1.21 +  Augmenting path algorithm for calculating maximum weighted
    1.22 +  perfect fractional matching in general graphs.
    1.23  
    1.24  \image html bipartite_matching.png
    1.25  \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
     2.1 --- a/lemon/Makefile.am	Sun Sep 20 21:38:24 2009 +0200
     2.2 +++ b/lemon/Makefile.am	Fri Sep 25 21:51:36 2009 +0200
     2.3 @@ -81,6 +81,7 @@
     2.4  	lemon/euler.h \
     2.5  	lemon/fib_heap.h \
     2.6  	lemon/fourary_heap.h \
     2.7 +	lemon/fractional_matching.h \
     2.8  	lemon/full_graph.h \
     2.9  	lemon/glpk.h \
    2.10  	lemon/gomory_hu.h \
     3.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     3.2 +++ b/lemon/fractional_matching.h	Fri Sep 25 21:51:36 2009 +0200
     3.3 @@ -0,0 +1,2135 @@
     3.4 +/* -*- mode: C++; indent-tabs-mode: nil; -*-
     3.5 + *
     3.6 + * This file is a part of LEMON, a generic C++ optimization library.
     3.7 + *
     3.8 + * Copyright (C) 2003-2009
     3.9 + * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    3.10 + * (Egervary Research Group on Combinatorial Optimization, EGRES).
    3.11 + *
    3.12 + * Permission to use, modify and distribute this software is granted
    3.13 + * provided that this copyright notice appears in all copies. For
    3.14 + * precise terms see the accompanying LICENSE file.
    3.15 + *
    3.16 + * This software is provided "AS IS" with no warranty of any kind,
    3.17 + * express or implied, and with no claim as to its suitability for any
    3.18 + * purpose.
    3.19 + *
    3.20 + */
    3.21 +
    3.22 +#ifndef LEMON_FRACTIONAL_MATCHING_H
    3.23 +#define LEMON_FRACTIONAL_MATCHING_H
    3.24 +
    3.25 +#include <vector>
    3.26 +#include <queue>
    3.27 +#include <set>
    3.28 +#include <limits>
    3.29 +
    3.30 +#include <lemon/core.h>
    3.31 +#include <lemon/unionfind.h>
    3.32 +#include <lemon/bin_heap.h>
    3.33 +#include <lemon/maps.h>
    3.34 +#include <lemon/assert.h>
    3.35 +#include <lemon/elevator.h>
    3.36 +
    3.37 +///\ingroup matching
    3.38 +///\file
    3.39 +///\brief Fractional matching algorithms in general graphs.
    3.40 +
    3.41 +namespace lemon {
    3.42 +
    3.43 +  /// \brief Default traits class of MaxFractionalMatching class.
    3.44 +  ///
    3.45 +  /// Default traits class of MaxFractionalMatching class.
    3.46 +  /// \tparam GR Graph type.
    3.47 +  template <typename GR>
    3.48 +  struct MaxFractionalMatchingDefaultTraits {
    3.49 +
    3.50 +    /// \brief The type of the graph the algorithm runs on.
    3.51 +    typedef GR Graph;
    3.52 +
    3.53 +    /// \brief The type of the map that stores the matching.
    3.54 +    ///
    3.55 +    /// The type of the map that stores the matching arcs.
    3.56 +    /// It must meet the \ref concepts::ReadWriteMap "ReadWriteMap" concept.
    3.57 +    typedef typename Graph::template NodeMap<typename GR::Arc> MatchingMap;
    3.58 +
    3.59 +    /// \brief Instantiates a MatchingMap.
    3.60 +    ///
    3.61 +    /// This function instantiates a \ref MatchingMap.
    3.62 +    /// \param graph The graph for which we would like to define
    3.63 +    /// the matching map.
    3.64 +    static MatchingMap* createMatchingMap(const Graph& graph) {
    3.65 +      return new MatchingMap(graph);
    3.66 +    }
    3.67 +
    3.68 +    /// \brief The elevator type used by MaxFractionalMatching algorithm.
    3.69 +    ///
    3.70 +    /// The elevator type used by MaxFractionalMatching algorithm.
    3.71 +    ///
    3.72 +    /// \sa Elevator
    3.73 +    /// \sa LinkedElevator
    3.74 +    typedef LinkedElevator<Graph, typename Graph::Node> Elevator;
    3.75 +
    3.76 +    /// \brief Instantiates an Elevator.
    3.77 +    ///
    3.78 +    /// This function instantiates an \ref Elevator.
    3.79 +    /// \param graph The graph for which we would like to define
    3.80 +    /// the elevator.
    3.81 +    /// \param max_level The maximum level of the elevator.
    3.82 +    static Elevator* createElevator(const Graph& graph, int max_level) {
    3.83 +      return new Elevator(graph, max_level);
    3.84 +    }
    3.85 +  };
    3.86 +
    3.87 +  /// \ingroup matching
    3.88 +  ///
    3.89 +  /// \brief Max cardinality fractional matching
    3.90 +  ///
    3.91 +  /// This class provides an implementation of fractional matching
    3.92 +  /// algorithm based on push-relabel principle.
    3.93 +  ///
    3.94 +  /// The maximum cardinality fractional matching is a relaxation of the
    3.95 +  /// maximum cardinality matching problem where the odd set constraints
    3.96 +  /// are omitted.
    3.97 +  /// It can be formulated with the following linear program.
    3.98 +  /// \f[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\f]
    3.99 +  /// \f[x_e \ge 0\quad \forall e\in E\f]
   3.100 +  /// \f[\max \sum_{e\in E}x_e\f]
   3.101 +  /// where \f$\delta(X)\f$ is the set of edges incident to a node in
   3.102 +  /// \f$X\f$. The result can be represented as the union of a
   3.103 +  /// matching with one value edges and a set of odd length cycles
   3.104 +  /// with half value edges.
   3.105 +  ///
   3.106 +  /// The algorithm calculates an optimal fractional matching and a
   3.107 +  /// barrier. The number of adjacents of any node set minus the size
   3.108 +  /// of node set is a lower bound on the uncovered nodes in the
   3.109 +  /// graph. For maximum matching a barrier is computed which
   3.110 +  /// maximizes this difference.
   3.111 +  ///
   3.112 +  /// The algorithm can be executed with the run() function.  After it
   3.113 +  /// the matching (the primal solution) and the barrier (the dual
   3.114 +  /// solution) can be obtained using the query functions.
   3.115 +  ///
   3.116 +  /// The primal solution is multiplied by
   3.117 +  /// \ref MaxWeightedMatching::primalScale "2".
   3.118 +  ///
   3.119 +  /// \tparam GR The undirected graph type the algorithm runs on.
   3.120 +#ifdef DOXYGEN
   3.121 +  template <typename GR, typename TR>
   3.122 +#else
   3.123 +  template <typename GR,
   3.124 +            typename TR = MaxFractionalMatchingDefaultTraits<GR> >
   3.125 +#endif
   3.126 +  class MaxFractionalMatching {
   3.127 +  public:
   3.128 +
   3.129 +    /// \brief The \ref MaxFractionalMatchingDefaultTraits "traits
   3.130 +    /// class" of the algorithm.
   3.131 +    typedef TR Traits;
   3.132 +    /// The type of the graph the algorithm runs on.
   3.133 +    typedef typename TR::Graph Graph;
   3.134 +    /// The type of the matching map.
   3.135 +    typedef typename TR::MatchingMap MatchingMap;
   3.136 +    /// The type of the elevator.
   3.137 +    typedef typename TR::Elevator Elevator;
   3.138 +
   3.139 +    /// \brief Scaling factor for primal solution
   3.140 +    ///
   3.141 +    /// Scaling factor for primal solution.
   3.142 +    static const int primalScale = 2;
   3.143 +
   3.144 +  private:
   3.145 +
   3.146 +    const Graph &_graph;
   3.147 +    int _node_num;
   3.148 +    bool _allow_loops;
   3.149 +    int _empty_level;
   3.150 +
   3.151 +    TEMPLATE_GRAPH_TYPEDEFS(Graph);
   3.152 +
   3.153 +    bool _local_matching;
   3.154 +    MatchingMap *_matching;
   3.155 +
   3.156 +    bool _local_level;
   3.157 +    Elevator *_level;
   3.158 +
   3.159 +    typedef typename Graph::template NodeMap<int> InDegMap;
   3.160 +    InDegMap *_indeg;
   3.161 +
   3.162 +    void createStructures() {
   3.163 +      _node_num = countNodes(_graph);
   3.164 +
   3.165 +      if (!_matching) {
   3.166 +        _local_matching = true;
   3.167 +        _matching = Traits::createMatchingMap(_graph);
   3.168 +      }
   3.169 +      if (!_level) {
   3.170 +        _local_level = true;
   3.171 +        _level = Traits::createElevator(_graph, _node_num);
   3.172 +      }
   3.173 +      if (!_indeg) {
   3.174 +        _indeg = new InDegMap(_graph);
   3.175 +      }
   3.176 +    }
   3.177 +
   3.178 +    void destroyStructures() {
   3.179 +      if (_local_matching) {
   3.180 +        delete _matching;
   3.181 +      }
   3.182 +      if (_local_level) {
   3.183 +        delete _level;
   3.184 +      }
   3.185 +      if (_indeg) {
   3.186 +        delete _indeg;
   3.187 +      }
   3.188 +    }
   3.189 +
   3.190 +    void postprocessing() {
   3.191 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.192 +        if ((*_indeg)[n] != 0) continue;
   3.193 +        _indeg->set(n, -1);
   3.194 +        Node u = n;
   3.195 +        while ((*_matching)[u] != INVALID) {
   3.196 +          Node v = _graph.target((*_matching)[u]);
   3.197 +          _indeg->set(v, -1);
   3.198 +          Arc a = _graph.oppositeArc((*_matching)[u]);
   3.199 +          u = _graph.target((*_matching)[v]);
   3.200 +          _indeg->set(u, -1);
   3.201 +          _matching->set(v, a);
   3.202 +        }
   3.203 +      }
   3.204 +
   3.205 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.206 +        if ((*_indeg)[n] != 1) continue;
   3.207 +        _indeg->set(n, -1);
   3.208 +
   3.209 +        int num = 1;
   3.210 +        Node u = _graph.target((*_matching)[n]);
   3.211 +        while (u != n) {
   3.212 +          _indeg->set(u, -1);
   3.213 +          u = _graph.target((*_matching)[u]);
   3.214 +          ++num;
   3.215 +        }
   3.216 +        if (num % 2 == 0 && num > 2) {
   3.217 +          Arc prev = _graph.oppositeArc((*_matching)[n]);
   3.218 +          Node v = _graph.target((*_matching)[n]);
   3.219 +          u = _graph.target((*_matching)[v]);
   3.220 +          _matching->set(v, prev);
   3.221 +          while (u != n) {
   3.222 +            prev = _graph.oppositeArc((*_matching)[u]);
   3.223 +            v = _graph.target((*_matching)[u]);
   3.224 +            u = _graph.target((*_matching)[v]);
   3.225 +            _matching->set(v, prev);
   3.226 +          }
   3.227 +        }
   3.228 +      }
   3.229 +    }
   3.230 +
   3.231 +  public:
   3.232 +
   3.233 +    typedef MaxFractionalMatching Create;
   3.234 +
   3.235 +    ///\name Named Template Parameters
   3.236 +
   3.237 +    ///@{
   3.238 +
   3.239 +    template <typename T>
   3.240 +    struct SetMatchingMapTraits : public Traits {
   3.241 +      typedef T MatchingMap;
   3.242 +      static MatchingMap *createMatchingMap(const Graph&) {
   3.243 +        LEMON_ASSERT(false, "MatchingMap is not initialized");
   3.244 +        return 0; // ignore warnings
   3.245 +      }
   3.246 +    };
   3.247 +
   3.248 +    /// \brief \ref named-templ-param "Named parameter" for setting
   3.249 +    /// MatchingMap type
   3.250 +    ///
   3.251 +    /// \ref named-templ-param "Named parameter" for setting MatchingMap
   3.252 +    /// type.
   3.253 +    template <typename T>
   3.254 +    struct SetMatchingMap
   3.255 +      : public MaxFractionalMatching<Graph, SetMatchingMapTraits<T> > {
   3.256 +      typedef MaxFractionalMatching<Graph, SetMatchingMapTraits<T> > Create;
   3.257 +    };
   3.258 +
   3.259 +    template <typename T>
   3.260 +    struct SetElevatorTraits : public Traits {
   3.261 +      typedef T Elevator;
   3.262 +      static Elevator *createElevator(const Graph&, int) {
   3.263 +        LEMON_ASSERT(false, "Elevator is not initialized");
   3.264 +        return 0; // ignore warnings
   3.265 +      }
   3.266 +    };
   3.267 +
   3.268 +    /// \brief \ref named-templ-param "Named parameter" for setting
   3.269 +    /// Elevator type
   3.270 +    ///
   3.271 +    /// \ref named-templ-param "Named parameter" for setting Elevator
   3.272 +    /// type. If this named parameter is used, then an external
   3.273 +    /// elevator object must be passed to the algorithm using the
   3.274 +    /// \ref elevator(Elevator&) "elevator()" function before calling
   3.275 +    /// \ref run() or \ref init().
   3.276 +    /// \sa SetStandardElevator
   3.277 +    template <typename T>
   3.278 +    struct SetElevator
   3.279 +      : public MaxFractionalMatching<Graph, SetElevatorTraits<T> > {
   3.280 +      typedef MaxFractionalMatching<Graph, SetElevatorTraits<T> > Create;
   3.281 +    };
   3.282 +
   3.283 +    template <typename T>
   3.284 +    struct SetStandardElevatorTraits : public Traits {
   3.285 +      typedef T Elevator;
   3.286 +      static Elevator *createElevator(const Graph& graph, int max_level) {
   3.287 +        return new Elevator(graph, max_level);
   3.288 +      }
   3.289 +    };
   3.290 +
   3.291 +    /// \brief \ref named-templ-param "Named parameter" for setting
   3.292 +    /// Elevator type with automatic allocation
   3.293 +    ///
   3.294 +    /// \ref named-templ-param "Named parameter" for setting Elevator
   3.295 +    /// type with automatic allocation.
   3.296 +    /// The Elevator should have standard constructor interface to be
   3.297 +    /// able to automatically created by the algorithm (i.e. the
   3.298 +    /// graph and the maximum level should be passed to it).
   3.299 +    /// However an external elevator object could also be passed to the
   3.300 +    /// algorithm with the \ref elevator(Elevator&) "elevator()" function
   3.301 +    /// before calling \ref run() or \ref init().
   3.302 +    /// \sa SetElevator
   3.303 +    template <typename T>
   3.304 +    struct SetStandardElevator
   3.305 +      : public MaxFractionalMatching<Graph, SetStandardElevatorTraits<T> > {
   3.306 +      typedef MaxFractionalMatching<Graph,
   3.307 +                                    SetStandardElevatorTraits<T> > Create;
   3.308 +    };
   3.309 +
   3.310 +    /// @}
   3.311 +
   3.312 +  protected:
   3.313 +
   3.314 +    MaxFractionalMatching() {}
   3.315 +
   3.316 +  public:
   3.317 +
   3.318 +    /// \brief Constructor
   3.319 +    ///
   3.320 +    /// Constructor.
   3.321 +    ///
   3.322 +    MaxFractionalMatching(const Graph &graph, bool allow_loops = true)
   3.323 +      : _graph(graph), _allow_loops(allow_loops),
   3.324 +        _local_matching(false), _matching(0),
   3.325 +        _local_level(false), _level(0),  _indeg(0)
   3.326 +    {}
   3.327 +
   3.328 +    ~MaxFractionalMatching() {
   3.329 +      destroyStructures();
   3.330 +    }
   3.331 +
   3.332 +    /// \brief Sets the matching map.
   3.333 +    ///
   3.334 +    /// Sets the matching map.
   3.335 +    /// If you don't use this function before calling \ref run() or
   3.336 +    /// \ref init(), an instance will be allocated automatically.
   3.337 +    /// The destructor deallocates this automatically allocated map,
   3.338 +    /// of course.
   3.339 +    /// \return <tt>(*this)</tt>
   3.340 +    MaxFractionalMatching& matchingMap(MatchingMap& map) {
   3.341 +      if (_local_matching) {
   3.342 +        delete _matching;
   3.343 +        _local_matching = false;
   3.344 +      }
   3.345 +      _matching = &map;
   3.346 +      return *this;
   3.347 +    }
   3.348 +
   3.349 +    /// \brief Sets the elevator used by algorithm.
   3.350 +    ///
   3.351 +    /// Sets the elevator used by algorithm.
   3.352 +    /// If you don't use this function before calling \ref run() or
   3.353 +    /// \ref init(), an instance will be allocated automatically.
   3.354 +    /// The destructor deallocates this automatically allocated elevator,
   3.355 +    /// of course.
   3.356 +    /// \return <tt>(*this)</tt>
   3.357 +    MaxFractionalMatching& elevator(Elevator& elevator) {
   3.358 +      if (_local_level) {
   3.359 +        delete _level;
   3.360 +        _local_level = false;
   3.361 +      }
   3.362 +      _level = &elevator;
   3.363 +      return *this;
   3.364 +    }
   3.365 +
   3.366 +    /// \brief Returns a const reference to the elevator.
   3.367 +    ///
   3.368 +    /// Returns a const reference to the elevator.
   3.369 +    ///
   3.370 +    /// \pre Either \ref run() or \ref init() must be called before
   3.371 +    /// using this function.
   3.372 +    const Elevator& elevator() const {
   3.373 +      return *_level;
   3.374 +    }
   3.375 +
   3.376 +    /// \name Execution control
   3.377 +    /// The simplest way to execute the algorithm is to use one of the
   3.378 +    /// member functions called \c run(). \n
   3.379 +    /// If you need more control on the execution, first
   3.380 +    /// you must call \ref init() and then one variant of the start()
   3.381 +    /// member.
   3.382 +
   3.383 +    /// @{
   3.384 +
   3.385 +    /// \brief Initializes the internal data structures.
   3.386 +    ///
   3.387 +    /// Initializes the internal data structures and sets the initial
   3.388 +    /// matching.
   3.389 +    void init() {
   3.390 +      createStructures();
   3.391 +
   3.392 +      _level->initStart();
   3.393 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.394 +        _indeg->set(n, 0);
   3.395 +        _matching->set(n, INVALID);
   3.396 +        _level->initAddItem(n);
   3.397 +      }
   3.398 +      _level->initFinish();
   3.399 +
   3.400 +      _empty_level = _node_num;
   3.401 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.402 +        for (OutArcIt a(_graph, n); a != INVALID; ++a) {
   3.403 +          if (_graph.target(a) == n && !_allow_loops) continue;
   3.404 +          _matching->set(n, a);
   3.405 +          Node v = _graph.target((*_matching)[n]);
   3.406 +          _indeg->set(v, (*_indeg)[v] + 1);
   3.407 +          break;
   3.408 +        }
   3.409 +      }
   3.410 +
   3.411 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.412 +        if ((*_indeg)[n] == 0) {
   3.413 +          _level->activate(n);
   3.414 +        }
   3.415 +      }
   3.416 +    }
   3.417 +
   3.418 +    /// \brief Starts the algorithm and computes a fractional matching
   3.419 +    ///
   3.420 +    /// The algorithm computes a maximum fractional matching.
   3.421 +    ///
   3.422 +    /// \param postprocess The algorithm computes first a matching
   3.423 +    /// which is a union of a matching with one value edges, cycles
   3.424 +    /// with half value edges and even length paths with half value
   3.425 +    /// edges. If the parameter is true, then after the push-relabel
   3.426 +    /// algorithm it postprocesses the matching to contain only
   3.427 +    /// matching edges and half value odd cycles.
   3.428 +    void start(bool postprocess = true) {
   3.429 +      Node n;
   3.430 +      while ((n = _level->highestActive()) != INVALID) {
   3.431 +        int level = _level->highestActiveLevel();
   3.432 +        int new_level = _level->maxLevel();
   3.433 +        for (InArcIt a(_graph, n); a != INVALID; ++a) {
   3.434 +          Node u = _graph.source(a);
   3.435 +          if (n == u && !_allow_loops) continue;
   3.436 +          Node v = _graph.target((*_matching)[u]);
   3.437 +          if ((*_level)[v] < level) {
   3.438 +            _indeg->set(v, (*_indeg)[v] - 1);
   3.439 +            if ((*_indeg)[v] == 0) {
   3.440 +              _level->activate(v);
   3.441 +            }
   3.442 +            _matching->set(u, a);
   3.443 +            _indeg->set(n, (*_indeg)[n] + 1);
   3.444 +            _level->deactivate(n);
   3.445 +            goto no_more_push;
   3.446 +          } else if (new_level > (*_level)[v]) {
   3.447 +            new_level = (*_level)[v];
   3.448 +          }
   3.449 +        }
   3.450 +
   3.451 +        if (new_level + 1 < _level->maxLevel()) {
   3.452 +          _level->liftHighestActive(new_level + 1);
   3.453 +        } else {
   3.454 +          _level->liftHighestActiveToTop();
   3.455 +        }
   3.456 +        if (_level->emptyLevel(level)) {
   3.457 +          _level->liftToTop(level);
   3.458 +        }
   3.459 +      no_more_push:
   3.460 +        ;
   3.461 +      }
   3.462 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.463 +        if ((*_matching)[n] == INVALID) continue;
   3.464 +        Node u = _graph.target((*_matching)[n]);
   3.465 +        if ((*_indeg)[u] > 1) {
   3.466 +          _indeg->set(u, (*_indeg)[u] - 1);
   3.467 +          _matching->set(n, INVALID);
   3.468 +        }
   3.469 +      }
   3.470 +      if (postprocess) {
   3.471 +        postprocessing();
   3.472 +      }
   3.473 +    }
   3.474 +
   3.475 +    /// \brief Starts the algorithm and computes a perfect fractional
   3.476 +    /// matching
   3.477 +    ///
   3.478 +    /// The algorithm computes a perfect fractional matching. If it
   3.479 +    /// does not exists, then the algorithm returns false and the
   3.480 +    /// matching is undefined and the barrier.
   3.481 +    ///
   3.482 +    /// \param postprocess The algorithm computes first a matching
   3.483 +    /// which is a union of a matching with one value edges, cycles
   3.484 +    /// with half value edges and even length paths with half value
   3.485 +    /// edges. If the parameter is true, then after the push-relabel
   3.486 +    /// algorithm it postprocesses the matching to contain only
   3.487 +    /// matching edges and half value odd cycles.
   3.488 +    bool startPerfect(bool postprocess = true) {
   3.489 +      Node n;
   3.490 +      while ((n = _level->highestActive()) != INVALID) {
   3.491 +        int level = _level->highestActiveLevel();
   3.492 +        int new_level = _level->maxLevel();
   3.493 +        for (InArcIt a(_graph, n); a != INVALID; ++a) {
   3.494 +          Node u = _graph.source(a);
   3.495 +          if (n == u && !_allow_loops) continue;
   3.496 +          Node v = _graph.target((*_matching)[u]);
   3.497 +          if ((*_level)[v] < level) {
   3.498 +            _indeg->set(v, (*_indeg)[v] - 1);
   3.499 +            if ((*_indeg)[v] == 0) {
   3.500 +              _level->activate(v);
   3.501 +            }
   3.502 +            _matching->set(u, a);
   3.503 +            _indeg->set(n, (*_indeg)[n] + 1);
   3.504 +            _level->deactivate(n);
   3.505 +            goto no_more_push;
   3.506 +          } else if (new_level > (*_level)[v]) {
   3.507 +            new_level = (*_level)[v];
   3.508 +          }
   3.509 +        }
   3.510 +
   3.511 +        if (new_level + 1 < _level->maxLevel()) {
   3.512 +          _level->liftHighestActive(new_level + 1);
   3.513 +        } else {
   3.514 +          _level->liftHighestActiveToTop();
   3.515 +          _empty_level = _level->maxLevel() - 1;
   3.516 +          return false;
   3.517 +        }
   3.518 +        if (_level->emptyLevel(level)) {
   3.519 +          _level->liftToTop(level);
   3.520 +          _empty_level = level;
   3.521 +          return false;
   3.522 +        }
   3.523 +      no_more_push:
   3.524 +        ;
   3.525 +      }
   3.526 +      if (postprocess) {
   3.527 +        postprocessing();
   3.528 +      }
   3.529 +      return true;
   3.530 +    }
   3.531 +
   3.532 +    /// \brief Runs the algorithm
   3.533 +    ///
   3.534 +    /// Just a shortcut for the next code:
   3.535 +    ///\code
   3.536 +    /// init();
   3.537 +    /// start();
   3.538 +    ///\endcode
   3.539 +    void run(bool postprocess = true) {
   3.540 +      init();
   3.541 +      start(postprocess);
   3.542 +    }
   3.543 +
   3.544 +    /// \brief Runs the algorithm to find a perfect fractional matching
   3.545 +    ///
   3.546 +    /// Just a shortcut for the next code:
   3.547 +    ///\code
   3.548 +    /// init();
   3.549 +    /// startPerfect();
   3.550 +    ///\endcode
   3.551 +    bool runPerfect(bool postprocess = true) {
   3.552 +      init();
   3.553 +      return startPerfect(postprocess);
   3.554 +    }
   3.555 +
   3.556 +    ///@}
   3.557 +
   3.558 +    /// \name Query Functions
   3.559 +    /// The result of the %Matching algorithm can be obtained using these
   3.560 +    /// functions.\n
   3.561 +    /// Before the use of these functions,
   3.562 +    /// either run() or start() must be called.
   3.563 +    ///@{
   3.564 +
   3.565 +
   3.566 +    /// \brief Return the number of covered nodes in the matching.
   3.567 +    ///
   3.568 +    /// This function returns the number of covered nodes in the matching.
   3.569 +    ///
   3.570 +    /// \pre Either run() or start() must be called before using this function.
   3.571 +    int matchingSize() const {
   3.572 +      int num = 0;
   3.573 +      for (NodeIt n(_graph); n != INVALID; ++n) {
   3.574 +        if ((*_matching)[n] != INVALID) {
   3.575 +          ++num;
   3.576 +        }
   3.577 +      }
   3.578 +      return num;
   3.579 +    }
   3.580 +
   3.581 +    /// \brief Returns a const reference to the matching map.
   3.582 +    ///
   3.583 +    /// Returns a const reference to the node map storing the found
   3.584 +    /// fractional matching. This method can be called after
   3.585 +    /// running the algorithm.
   3.586 +    ///
   3.587 +    /// \pre Either \ref run() or \ref init() must be called before
   3.588 +    /// using this function.
   3.589 +    const MatchingMap& matchingMap() const {
   3.590 +      return *_matching;
   3.591 +    }
   3.592 +
   3.593 +    /// \brief Return \c true if the given edge is in the matching.
   3.594 +    ///
   3.595 +    /// This function returns \c true if the given edge is in the
   3.596 +    /// found matching. The result is scaled by \ref primalScale
   3.597 +    /// "primal scale".
   3.598 +    ///
   3.599 +    /// \pre Either run() or start() must be called before using this function.
   3.600 +    int matching(const Edge& edge) const {
   3.601 +      return (edge == (*_matching)[_graph.u(edge)] ? 1 : 0) +
   3.602 +        (edge == (*_matching)[_graph.v(edge)] ? 1 : 0);
   3.603 +    }
   3.604 +
   3.605 +    /// \brief Return the fractional matching arc (or edge) incident
   3.606 +    /// to the given node.
   3.607 +    ///
   3.608 +    /// This function returns one of the fractional matching arc (or
   3.609 +    /// edge) incident to the given node in the found matching or \c
   3.610 +    /// INVALID if the node is not covered by the matching or if the
   3.611 +    /// node is on an odd length cycle then it is the successor edge
   3.612 +    /// on the cycle.
   3.613 +    ///
   3.614 +    /// \pre Either run() or start() must be called before using this function.
   3.615 +    Arc matching(const Node& node) const {
   3.616 +      return (*_matching)[node];
   3.617 +    }
   3.618 +
   3.619 +    /// \brief Returns true if the node is in the barrier
   3.620 +    ///
   3.621 +    /// The barrier is a subset of the nodes. If the nodes in the
   3.622 +    /// barrier have less adjacent nodes than the size of the barrier,
   3.623 +    /// then at least as much nodes cannot be covered as the
   3.624 +    /// difference of the two subsets.
   3.625 +    bool barrier(const Node& node) const {
   3.626 +      return (*_level)[node] >= _empty_level;
   3.627 +    }
   3.628 +
   3.629 +    /// @}
   3.630 +
   3.631 +  };
   3.632 +
   3.633 +  /// \ingroup matching
   3.634 +  ///
   3.635 +  /// \brief Weighted fractional matching in general graphs
   3.636 +  ///
   3.637 +  /// This class provides an efficient implementation of fractional
   3.638 +  /// matching algorithm. The implementation is based on extensive use
   3.639 +  /// of priority queues and provides \f$O(nm\log n)\f$ time
   3.640 +  /// complexity.
   3.641 +  ///
   3.642 +  /// The maximum weighted fractional matching is a relaxation of the
   3.643 +  /// maximum weighted matching problem where the odd set constraints
   3.644 +  /// are omitted.
   3.645 +  /// It can be formulated with the following linear program.
   3.646 +  /// \f[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\f]
   3.647 +  /// \f[x_e \ge 0\quad \forall e\in E\f]
   3.648 +  /// \f[\max \sum_{e\in E}x_ew_e\f]
   3.649 +  /// where \f$\delta(X)\f$ is the set of edges incident to a node in
   3.650 +  /// \f$X\f$. The result must be the union of a matching with one
   3.651 +  /// value edges and a set of odd length cycles with half value edges.
   3.652 +  ///
   3.653 +  /// The algorithm calculates an optimal fractional matching and a
   3.654 +  /// proof of the optimality. The solution of the dual problem can be
   3.655 +  /// used to check the result of the algorithm. The dual linear
   3.656 +  /// problem is the following.
   3.657 +  /// \f[ y_u + y_v \ge w_{uv} \quad \forall uv\in E\f]
   3.658 +  /// \f[y_u \ge 0 \quad \forall u \in V\f]
   3.659 +  /// \f[\min \sum_{u \in V}y_u \f] ///
   3.660 +  ///
   3.661 +  /// The algorithm can be executed with the run() function.
   3.662 +  /// After it the matching (the primal solution) and the dual solution
   3.663 +  /// can be obtained using the query functions.
   3.664 +  ///
   3.665 +  /// If the value type is integer, then the primal and the dual
   3.666 +  /// solutions are multiplied by
   3.667 +  /// \ref MaxWeightedMatching::primalScale "2" and
   3.668 +  /// \ref MaxWeightedMatching::dualScale "4" respectively.
   3.669 +  ///
   3.670 +  /// \tparam GR The undirected graph type the algorithm runs on.
   3.671 +  /// \tparam WM The type edge weight map. The default type is
   3.672 +  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".
   3.673 +#ifdef DOXYGEN
   3.674 +  template <typename GR, typename WM>
   3.675 +#else
   3.676 +  template <typename GR,
   3.677 +            typename WM = typename GR::template EdgeMap<int> >
   3.678 +#endif
   3.679 +  class MaxWeightedFractionalMatching {
   3.680 +  public:
   3.681 +
   3.682 +    /// The graph type of the algorithm
   3.683 +    typedef GR Graph;
   3.684 +    /// The type of the edge weight map
   3.685 +    typedef WM WeightMap;
   3.686 +    /// The value type of the edge weights
   3.687 +    typedef typename WeightMap::Value Value;
   3.688 +
   3.689 +    /// The type of the matching map
   3.690 +    typedef typename Graph::template NodeMap<typename Graph::Arc>
   3.691 +    MatchingMap;
   3.692 +
   3.693 +    /// \brief Scaling factor for primal solution
   3.694 +    ///
   3.695 +    /// Scaling factor for primal solution. It is equal to 2 or 1
   3.696 +    /// according to the value type.
   3.697 +    static const int primalScale =
   3.698 +      std::numeric_limits<Value>::is_integer ? 2 : 1;
   3.699 +
   3.700 +    /// \brief Scaling factor for dual solution
   3.701 +    ///
   3.702 +    /// Scaling factor for dual solution. It is equal to 4 or 1
   3.703 +    /// according to the value type.
   3.704 +    static const int dualScale =
   3.705 +      std::numeric_limits<Value>::is_integer ? 4 : 1;
   3.706 +
   3.707 +  private:
   3.708 +
   3.709 +    TEMPLATE_GRAPH_TYPEDEFS(Graph);
   3.710 +
   3.711 +    typedef typename Graph::template NodeMap<Value> NodePotential;
   3.712 +
   3.713 +    const Graph& _graph;
   3.714 +    const WeightMap& _weight;
   3.715 +
   3.716 +    MatchingMap* _matching;
   3.717 +    NodePotential* _node_potential;
   3.718 +
   3.719 +    int _node_num;
   3.720 +    bool _allow_loops;
   3.721 +
   3.722 +    enum Status {
   3.723 +      EVEN = -1, MATCHED = 0, ODD = 1
   3.724 +    };
   3.725 +
   3.726 +    typedef typename Graph::template NodeMap<Status> StatusMap;
   3.727 +    StatusMap* _status;
   3.728 +
   3.729 +    typedef typename Graph::template NodeMap<Arc> PredMap;
   3.730 +    PredMap* _pred;
   3.731 +
   3.732 +    typedef ExtendFindEnum<IntNodeMap> TreeSet;
   3.733 +
   3.734 +    IntNodeMap *_tree_set_index;
   3.735 +    TreeSet *_tree_set;
   3.736 +
   3.737 +    IntNodeMap *_delta1_index;
   3.738 +    BinHeap<Value, IntNodeMap> *_delta1;
   3.739 +
   3.740 +    IntNodeMap *_delta2_index;
   3.741 +    BinHeap<Value, IntNodeMap> *_delta2;
   3.742 +
   3.743 +    IntEdgeMap *_delta3_index;
   3.744 +    BinHeap<Value, IntEdgeMap> *_delta3;
   3.745 +
   3.746 +    Value _delta_sum;
   3.747 +
   3.748 +    void createStructures() {
   3.749 +      _node_num = countNodes(_graph);
   3.750 +
   3.751 +      if (!_matching) {
   3.752 +        _matching = new MatchingMap(_graph);
   3.753 +      }
   3.754 +      if (!_node_potential) {
   3.755 +        _node_potential = new NodePotential(_graph);
   3.756 +      }
   3.757 +      if (!_status) {
   3.758 +        _status = new StatusMap(_graph);
   3.759 +      }
   3.760 +      if (!_pred) {
   3.761 +        _pred = new PredMap(_graph);
   3.762 +      }
   3.763 +      if (!_tree_set) {
   3.764 +        _tree_set_index = new IntNodeMap(_graph);
   3.765 +        _tree_set = new TreeSet(*_tree_set_index);
   3.766 +      }
   3.767 +      if (!_delta1) {
   3.768 +        _delta1_index = new IntNodeMap(_graph);
   3.769 +        _delta1 = new BinHeap<Value, IntNodeMap>(*_delta1_index);
   3.770 +      }
   3.771 +      if (!_delta2) {
   3.772 +        _delta2_index = new IntNodeMap(_graph);
   3.773 +        _delta2 = new BinHeap<Value, IntNodeMap>(*_delta2_index);
   3.774 +      }
   3.775 +      if (!_delta3) {
   3.776 +        _delta3_index = new IntEdgeMap(_graph);
   3.777 +        _delta3 = new BinHeap<Value, IntEdgeMap>(*_delta3_index);
   3.778 +      }
   3.779 +    }
   3.780 +
   3.781 +    void destroyStructures() {
   3.782 +      if (_matching) {
   3.783 +        delete _matching;
   3.784 +      }
   3.785 +      if (_node_potential) {
   3.786 +        delete _node_potential;
   3.787 +      }
   3.788 +      if (_status) {
   3.789 +        delete _status;
   3.790 +      }
   3.791 +      if (_pred) {
   3.792 +        delete _pred;
   3.793 +      }
   3.794 +      if (_tree_set) {
   3.795 +        delete _tree_set_index;
   3.796 +        delete _tree_set;
   3.797 +      }
   3.798 +      if (_delta1) {
   3.799 +        delete _delta1_index;
   3.800 +        delete _delta1;
   3.801 +      }
   3.802 +      if (_delta2) {
   3.803 +        delete _delta2_index;
   3.804 +        delete _delta2;
   3.805 +      }
   3.806 +      if (_delta3) {
   3.807 +        delete _delta3_index;
   3.808 +        delete _delta3;
   3.809 +      }
   3.810 +    }
   3.811 +
   3.812 +    void matchedToEven(Node node, int tree) {
   3.813 +      _tree_set->insert(node, tree);
   3.814 +      _node_potential->set(node, (*_node_potential)[node] + _delta_sum);
   3.815 +      _delta1->push(node, (*_node_potential)[node]);
   3.816 +
   3.817 +      if (_delta2->state(node) == _delta2->IN_HEAP) {
   3.818 +        _delta2->erase(node);
   3.819 +      }
   3.820 +
   3.821 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
   3.822 +        Node v = _graph.source(a);
   3.823 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
   3.824 +          dualScale * _weight[a];
   3.825 +        if (node == v) {
   3.826 +          if (_allow_loops && _graph.direction(a)) {
   3.827 +            _delta3->push(a, rw / 2);
   3.828 +          }
   3.829 +        } else if ((*_status)[v] == EVEN) {
   3.830 +          _delta3->push(a, rw / 2);
   3.831 +        } else if ((*_status)[v] == MATCHED) {
   3.832 +          if (_delta2->state(v) != _delta2->IN_HEAP) {
   3.833 +            _pred->set(v, a);
   3.834 +            _delta2->push(v, rw);
   3.835 +          } else if ((*_delta2)[v] > rw) {
   3.836 +            _pred->set(v, a);
   3.837 +            _delta2->decrease(v, rw);
   3.838 +          }
   3.839 +        }
   3.840 +      }
   3.841 +    }
   3.842 +
   3.843 +    void matchedToOdd(Node node, int tree) {
   3.844 +      _tree_set->insert(node, tree);
   3.845 +      _node_potential->set(node, (*_node_potential)[node] - _delta_sum);
   3.846 +
   3.847 +      if (_delta2->state(node) == _delta2->IN_HEAP) {
   3.848 +        _delta2->erase(node);
   3.849 +      }
   3.850 +    }
   3.851 +
   3.852 +    void evenToMatched(Node node, int tree) {
   3.853 +      _delta1->erase(node);
   3.854 +      _node_potential->set(node, (*_node_potential)[node] - _delta_sum);
   3.855 +      Arc min = INVALID;
   3.856 +      Value minrw = std::numeric_limits<Value>::max();
   3.857 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
   3.858 +        Node v = _graph.source(a);
   3.859 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
   3.860 +          dualScale * _weight[a];
   3.861 +
   3.862 +        if (node == v) {
   3.863 +          if (_allow_loops && _graph.direction(a)) {
   3.864 +            _delta3->erase(a);
   3.865 +          }
   3.866 +        } else if ((*_status)[v] == EVEN) {
   3.867 +          _delta3->erase(a);
   3.868 +          if (minrw > rw) {
   3.869 +            min = _graph.oppositeArc(a);
   3.870 +            minrw = rw;
   3.871 +          }
   3.872 +        } else if ((*_status)[v]  == MATCHED) {
   3.873 +          if ((*_pred)[v] == a) {
   3.874 +            Arc mina = INVALID;
   3.875 +            Value minrwa = std::numeric_limits<Value>::max();
   3.876 +            for (OutArcIt aa(_graph, v); aa != INVALID; ++aa) {
   3.877 +              Node va = _graph.target(aa);
   3.878 +              if ((*_status)[va] != EVEN ||
   3.879 +                  _tree_set->find(va) == tree) continue;
   3.880 +              Value rwa = (*_node_potential)[v] + (*_node_potential)[va] -
   3.881 +                dualScale * _weight[aa];
   3.882 +              if (minrwa > rwa) {
   3.883 +                minrwa = rwa;
   3.884 +                mina = aa;
   3.885 +              }
   3.886 +            }
   3.887 +            if (mina != INVALID) {
   3.888 +              _pred->set(v, mina);
   3.889 +              _delta2->increase(v, minrwa);
   3.890 +            } else {
   3.891 +              _pred->set(v, INVALID);
   3.892 +              _delta2->erase(v);
   3.893 +            }
   3.894 +          }
   3.895 +        }
   3.896 +      }
   3.897 +      if (min != INVALID) {
   3.898 +        _pred->set(node, min);
   3.899 +        _delta2->push(node, minrw);
   3.900 +      } else {
   3.901 +        _pred->set(node, INVALID);
   3.902 +      }
   3.903 +    }
   3.904 +
   3.905 +    void oddToMatched(Node node) {
   3.906 +      _node_potential->set(node, (*_node_potential)[node] + _delta_sum);
   3.907 +      Arc min = INVALID;
   3.908 +      Value minrw = std::numeric_limits<Value>::max();
   3.909 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
   3.910 +        Node v = _graph.source(a);
   3.911 +        if ((*_status)[v] != EVEN) continue;
   3.912 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
   3.913 +          dualScale * _weight[a];
   3.914 +
   3.915 +        if (minrw > rw) {
   3.916 +          min = _graph.oppositeArc(a);
   3.917 +          minrw = rw;
   3.918 +        }
   3.919 +      }
   3.920 +      if (min != INVALID) {
   3.921 +        _pred->set(node, min);
   3.922 +        _delta2->push(node, minrw);
   3.923 +      } else {
   3.924 +        _pred->set(node, INVALID);
   3.925 +      }
   3.926 +    }
   3.927 +
   3.928 +    void alternatePath(Node even, int tree) {
   3.929 +      Node odd;
   3.930 +
   3.931 +      _status->set(even, MATCHED);
   3.932 +      evenToMatched(even, tree);
   3.933 +
   3.934 +      Arc prev = (*_matching)[even];
   3.935 +      while (prev != INVALID) {
   3.936 +        odd = _graph.target(prev);
   3.937 +        even = _graph.target((*_pred)[odd]);
   3.938 +        _matching->set(odd, (*_pred)[odd]);
   3.939 +        _status->set(odd, MATCHED);
   3.940 +        oddToMatched(odd);
   3.941 +
   3.942 +        prev = (*_matching)[even];
   3.943 +        _status->set(even, MATCHED);
   3.944 +        _matching->set(even, _graph.oppositeArc((*_matching)[odd]));
   3.945 +        evenToMatched(even, tree);
   3.946 +      }
   3.947 +    }
   3.948 +
   3.949 +    void destroyTree(int tree) {
   3.950 +      for (typename TreeSet::ItemIt n(*_tree_set, tree); n != INVALID; ++n) {
   3.951 +        if ((*_status)[n] == EVEN) {
   3.952 +          _status->set(n, MATCHED);
   3.953 +          evenToMatched(n, tree);
   3.954 +        } else if ((*_status)[n] == ODD) {
   3.955 +          _status->set(n, MATCHED);
   3.956 +          oddToMatched(n);
   3.957 +        }
   3.958 +      }
   3.959 +      _tree_set->eraseClass(tree);
   3.960 +    }
   3.961 +
   3.962 +
   3.963 +    void unmatchNode(const Node& node) {
   3.964 +      int tree = _tree_set->find(node);
   3.965 +
   3.966 +      alternatePath(node, tree);
   3.967 +      destroyTree(tree);
   3.968 +
   3.969 +      _matching->set(node, INVALID);
   3.970 +    }
   3.971 +
   3.972 +
   3.973 +    void augmentOnEdge(const Edge& edge) {
   3.974 +      Node left = _graph.u(edge);
   3.975 +      int left_tree = _tree_set->find(left);
   3.976 +
   3.977 +      alternatePath(left, left_tree);
   3.978 +      destroyTree(left_tree);
   3.979 +      _matching->set(left, _graph.direct(edge, true));
   3.980 +
   3.981 +      Node right = _graph.v(edge);
   3.982 +      int right_tree = _tree_set->find(right);
   3.983 +
   3.984 +      alternatePath(right, right_tree);
   3.985 +      destroyTree(right_tree);
   3.986 +      _matching->set(right, _graph.direct(edge, false));
   3.987 +    }
   3.988 +
   3.989 +    void augmentOnArc(const Arc& arc) {
   3.990 +      Node left = _graph.source(arc);
   3.991 +      _status->set(left, MATCHED);
   3.992 +      _matching->set(left, arc);
   3.993 +      _pred->set(left, arc);
   3.994 +
   3.995 +      Node right = _graph.target(arc);
   3.996 +      int right_tree = _tree_set->find(right);
   3.997 +
   3.998 +      alternatePath(right, right_tree);
   3.999 +      destroyTree(right_tree);
  3.1000 +      _matching->set(right, _graph.oppositeArc(arc));
  3.1001 +    }
  3.1002 +
  3.1003 +    void extendOnArc(const Arc& arc) {
  3.1004 +      Node base = _graph.target(arc);
  3.1005 +      int tree = _tree_set->find(base);
  3.1006 +
  3.1007 +      Node odd = _graph.source(arc);
  3.1008 +      _tree_set->insert(odd, tree);
  3.1009 +      _status->set(odd, ODD);
  3.1010 +      matchedToOdd(odd, tree);
  3.1011 +      _pred->set(odd, arc);
  3.1012 +
  3.1013 +      Node even = _graph.target((*_matching)[odd]);
  3.1014 +      _tree_set->insert(even, tree);
  3.1015 +      _status->set(even, EVEN);
  3.1016 +      matchedToEven(even, tree);
  3.1017 +    }
  3.1018 +
  3.1019 +    void cycleOnEdge(const Edge& edge, int tree) {
  3.1020 +      Node nca = INVALID;
  3.1021 +      std::vector<Node> left_path, right_path;
  3.1022 +
  3.1023 +      {
  3.1024 +        std::set<Node> left_set, right_set;
  3.1025 +        Node left = _graph.u(edge);
  3.1026 +        left_path.push_back(left);
  3.1027 +        left_set.insert(left);
  3.1028 +
  3.1029 +        Node right = _graph.v(edge);
  3.1030 +        right_path.push_back(right);
  3.1031 +        right_set.insert(right);
  3.1032 +
  3.1033 +        while (true) {
  3.1034 +
  3.1035 +          if (left_set.find(right) != left_set.end()) {
  3.1036 +            nca = right;
  3.1037 +            break;
  3.1038 +          }
  3.1039 +
  3.1040 +          if ((*_matching)[left] == INVALID) break;
  3.1041 +
  3.1042 +          left = _graph.target((*_matching)[left]);
  3.1043 +          left_path.push_back(left);
  3.1044 +          left = _graph.target((*_pred)[left]);
  3.1045 +          left_path.push_back(left);
  3.1046 +
  3.1047 +          left_set.insert(left);
  3.1048 +
  3.1049 +          if (right_set.find(left) != right_set.end()) {
  3.1050 +            nca = left;
  3.1051 +            break;
  3.1052 +          }
  3.1053 +
  3.1054 +          if ((*_matching)[right] == INVALID) break;
  3.1055 +
  3.1056 +          right = _graph.target((*_matching)[right]);
  3.1057 +          right_path.push_back(right);
  3.1058 +          right = _graph.target((*_pred)[right]);
  3.1059 +          right_path.push_back(right);
  3.1060 +
  3.1061 +          right_set.insert(right);
  3.1062 +
  3.1063 +        }
  3.1064 +
  3.1065 +        if (nca == INVALID) {
  3.1066 +          if ((*_matching)[left] == INVALID) {
  3.1067 +            nca = right;
  3.1068 +            while (left_set.find(nca) == left_set.end()) {
  3.1069 +              nca = _graph.target((*_matching)[nca]);
  3.1070 +              right_path.push_back(nca);
  3.1071 +              nca = _graph.target((*_pred)[nca]);
  3.1072 +              right_path.push_back(nca);
  3.1073 +            }
  3.1074 +          } else {
  3.1075 +            nca = left;
  3.1076 +            while (right_set.find(nca) == right_set.end()) {
  3.1077 +              nca = _graph.target((*_matching)[nca]);
  3.1078 +              left_path.push_back(nca);
  3.1079 +              nca = _graph.target((*_pred)[nca]);
  3.1080 +              left_path.push_back(nca);
  3.1081 +            }
  3.1082 +          }
  3.1083 +        }
  3.1084 +      }
  3.1085 +
  3.1086 +      alternatePath(nca, tree);
  3.1087 +      Arc prev;
  3.1088 +
  3.1089 +      prev = _graph.direct(edge, true);
  3.1090 +      for (int i = 0; left_path[i] != nca; i += 2) {
  3.1091 +        _matching->set(left_path[i], prev);
  3.1092 +        _status->set(left_path[i], MATCHED);
  3.1093 +        evenToMatched(left_path[i], tree);
  3.1094 +
  3.1095 +        prev = _graph.oppositeArc((*_pred)[left_path[i + 1]]);
  3.1096 +        _status->set(left_path[i + 1], MATCHED);
  3.1097 +        oddToMatched(left_path[i + 1]);
  3.1098 +      }
  3.1099 +      _matching->set(nca, prev);
  3.1100 +
  3.1101 +      for (int i = 0; right_path[i] != nca; i += 2) {
  3.1102 +        _status->set(right_path[i], MATCHED);
  3.1103 +        evenToMatched(right_path[i], tree);
  3.1104 +
  3.1105 +        _matching->set(right_path[i + 1], (*_pred)[right_path[i + 1]]);
  3.1106 +        _status->set(right_path[i + 1], MATCHED);
  3.1107 +        oddToMatched(right_path[i + 1]);
  3.1108 +      }
  3.1109 +
  3.1110 +      destroyTree(tree);
  3.1111 +    }
  3.1112 +
  3.1113 +    void extractCycle(const Arc &arc) {
  3.1114 +      Node left = _graph.source(arc);
  3.1115 +      Node odd = _graph.target((*_matching)[left]);
  3.1116 +      Arc prev;
  3.1117 +      while (odd != left) {
  3.1118 +        Node even = _graph.target((*_matching)[odd]);
  3.1119 +        prev = (*_matching)[odd];
  3.1120 +        odd = _graph.target((*_matching)[even]);
  3.1121 +        _matching->set(even, _graph.oppositeArc(prev));
  3.1122 +      }
  3.1123 +      _matching->set(left, arc);
  3.1124 +
  3.1125 +      Node right = _graph.target(arc);
  3.1126 +      int right_tree = _tree_set->find(right);
  3.1127 +      alternatePath(right, right_tree);
  3.1128 +      destroyTree(right_tree);
  3.1129 +      _matching->set(right, _graph.oppositeArc(arc));
  3.1130 +    }
  3.1131 +
  3.1132 +  public:
  3.1133 +
  3.1134 +    /// \brief Constructor
  3.1135 +    ///
  3.1136 +    /// Constructor.
  3.1137 +    MaxWeightedFractionalMatching(const Graph& graph, const WeightMap& weight,
  3.1138 +                                  bool allow_loops = true)
  3.1139 +      : _graph(graph), _weight(weight), _matching(0),
  3.1140 +      _node_potential(0), _node_num(0), _allow_loops(allow_loops),
  3.1141 +      _status(0),  _pred(0),
  3.1142 +      _tree_set_index(0), _tree_set(0),
  3.1143 +
  3.1144 +      _delta1_index(0), _delta1(0),
  3.1145 +      _delta2_index(0), _delta2(0),
  3.1146 +      _delta3_index(0), _delta3(0),
  3.1147 +
  3.1148 +      _delta_sum() {}
  3.1149 +
  3.1150 +    ~MaxWeightedFractionalMatching() {
  3.1151 +      destroyStructures();
  3.1152 +    }
  3.1153 +
  3.1154 +    /// \name Execution Control
  3.1155 +    /// The simplest way to execute the algorithm is to use the
  3.1156 +    /// \ref run() member function.
  3.1157 +
  3.1158 +    ///@{
  3.1159 +
  3.1160 +    /// \brief Initialize the algorithm
  3.1161 +    ///
  3.1162 +    /// This function initializes the algorithm.
  3.1163 +    void init() {
  3.1164 +      createStructures();
  3.1165 +
  3.1166 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1167 +        (*_delta1_index)[n] = _delta1->PRE_HEAP;
  3.1168 +        (*_delta2_index)[n] = _delta2->PRE_HEAP;
  3.1169 +      }
  3.1170 +      for (EdgeIt e(_graph); e != INVALID; ++e) {
  3.1171 +        (*_delta3_index)[e] = _delta3->PRE_HEAP;
  3.1172 +      }
  3.1173 +
  3.1174 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1175 +        Value max = 0;
  3.1176 +        for (OutArcIt e(_graph, n); e != INVALID; ++e) {
  3.1177 +          if (_graph.target(e) == n && !_allow_loops) continue;
  3.1178 +          if ((dualScale * _weight[e]) / 2 > max) {
  3.1179 +            max = (dualScale * _weight[e]) / 2;
  3.1180 +          }
  3.1181 +        }
  3.1182 +        _node_potential->set(n, max);
  3.1183 +        _delta1->push(n, max);
  3.1184 +
  3.1185 +        _tree_set->insert(n);
  3.1186 +
  3.1187 +        _matching->set(n, INVALID);
  3.1188 +        _status->set(n, EVEN);
  3.1189 +      }
  3.1190 +
  3.1191 +      for (EdgeIt e(_graph); e != INVALID; ++e) {
  3.1192 +        Node left = _graph.u(e);
  3.1193 +        Node right = _graph.v(e);
  3.1194 +        if (left == right && !_allow_loops) continue;
  3.1195 +        _delta3->push(e, ((*_node_potential)[left] +
  3.1196 +                          (*_node_potential)[right] -
  3.1197 +                          dualScale * _weight[e]) / 2);
  3.1198 +      }
  3.1199 +    }
  3.1200 +
  3.1201 +    /// \brief Start the algorithm
  3.1202 +    ///
  3.1203 +    /// This function starts the algorithm.
  3.1204 +    ///
  3.1205 +    /// \pre \ref init() must be called before using this function.
  3.1206 +    void start() {
  3.1207 +      enum OpType {
  3.1208 +        D1, D2, D3
  3.1209 +      };
  3.1210 +
  3.1211 +      int unmatched = _node_num;
  3.1212 +      while (unmatched > 0) {
  3.1213 +        Value d1 = !_delta1->empty() ?
  3.1214 +          _delta1->prio() : std::numeric_limits<Value>::max();
  3.1215 +
  3.1216 +        Value d2 = !_delta2->empty() ?
  3.1217 +          _delta2->prio() : std::numeric_limits<Value>::max();
  3.1218 +
  3.1219 +        Value d3 = !_delta3->empty() ?
  3.1220 +          _delta3->prio() : std::numeric_limits<Value>::max();
  3.1221 +
  3.1222 +        _delta_sum = d3; OpType ot = D3;
  3.1223 +        if (d1 < _delta_sum) { _delta_sum = d1; ot = D1; }
  3.1224 +        if (d2 < _delta_sum) { _delta_sum = d2; ot = D2; }
  3.1225 +
  3.1226 +        switch (ot) {
  3.1227 +        case D1:
  3.1228 +          {
  3.1229 +            Node n = _delta1->top();
  3.1230 +            unmatchNode(n);
  3.1231 +            --unmatched;
  3.1232 +          }
  3.1233 +          break;
  3.1234 +        case D2:
  3.1235 +          {
  3.1236 +            Node n = _delta2->top();
  3.1237 +            Arc a = (*_pred)[n];
  3.1238 +            if ((*_matching)[n] == INVALID) {
  3.1239 +              augmentOnArc(a);
  3.1240 +              --unmatched;
  3.1241 +            } else {
  3.1242 +              Node v = _graph.target((*_matching)[n]);
  3.1243 +              if ((*_matching)[n] !=
  3.1244 +                  _graph.oppositeArc((*_matching)[v])) {
  3.1245 +                extractCycle(a);
  3.1246 +                --unmatched;
  3.1247 +              } else {
  3.1248 +                extendOnArc(a);
  3.1249 +              }
  3.1250 +            }
  3.1251 +          } break;
  3.1252 +        case D3:
  3.1253 +          {
  3.1254 +            Edge e = _delta3->top();
  3.1255 +
  3.1256 +            Node left = _graph.u(e);
  3.1257 +            Node right = _graph.v(e);
  3.1258 +
  3.1259 +            int left_tree = _tree_set->find(left);
  3.1260 +            int right_tree = _tree_set->find(right);
  3.1261 +
  3.1262 +            if (left_tree == right_tree) {
  3.1263 +              cycleOnEdge(e, left_tree);
  3.1264 +              --unmatched;
  3.1265 +            } else {
  3.1266 +              augmentOnEdge(e);
  3.1267 +              unmatched -= 2;
  3.1268 +            }
  3.1269 +          } break;
  3.1270 +        }
  3.1271 +      }
  3.1272 +    }
  3.1273 +
  3.1274 +    /// \brief Run the algorithm.
  3.1275 +    ///
  3.1276 +    /// This method runs the \c %MaxWeightedMatching algorithm.
  3.1277 +    ///
  3.1278 +    /// \note mwfm.run() is just a shortcut of the following code.
  3.1279 +    /// \code
  3.1280 +    ///   mwfm.init();
  3.1281 +    ///   mwfm.start();
  3.1282 +    /// \endcode
  3.1283 +    void run() {
  3.1284 +      init();
  3.1285 +      start();
  3.1286 +    }
  3.1287 +
  3.1288 +    /// @}
  3.1289 +
  3.1290 +    /// \name Primal Solution
  3.1291 +    /// Functions to get the primal solution, i.e. the maximum weighted
  3.1292 +    /// matching.\n
  3.1293 +    /// Either \ref run() or \ref start() function should be called before
  3.1294 +    /// using them.
  3.1295 +
  3.1296 +    /// @{
  3.1297 +
  3.1298 +    /// \brief Return the weight of the matching.
  3.1299 +    ///
  3.1300 +    /// This function returns the weight of the found matching. This
  3.1301 +    /// value is scaled by \ref primalScale "primal scale".
  3.1302 +    ///
  3.1303 +    /// \pre Either run() or start() must be called before using this function.
  3.1304 +    Value matchingWeight() const {
  3.1305 +      Value sum = 0;
  3.1306 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1307 +        if ((*_matching)[n] != INVALID) {
  3.1308 +          sum += _weight[(*_matching)[n]];
  3.1309 +        }
  3.1310 +      }
  3.1311 +      return sum * primalScale / 2;
  3.1312 +    }
  3.1313 +
  3.1314 +    /// \brief Return the number of covered nodes in the matching.
  3.1315 +    ///
  3.1316 +    /// This function returns the number of covered nodes in the matching.
  3.1317 +    ///
  3.1318 +    /// \pre Either run() or start() must be called before using this function.
  3.1319 +    int matchingSize() const {
  3.1320 +      int num = 0;
  3.1321 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1322 +        if ((*_matching)[n] != INVALID) {
  3.1323 +          ++num;
  3.1324 +        }
  3.1325 +      }
  3.1326 +      return num;
  3.1327 +    }
  3.1328 +
  3.1329 +    /// \brief Return \c true if the given edge is in the matching.
  3.1330 +    ///
  3.1331 +    /// This function returns \c true if the given edge is in the
  3.1332 +    /// found matching. The result is scaled by \ref primalScale
  3.1333 +    /// "primal scale".
  3.1334 +    ///
  3.1335 +    /// \pre Either run() or start() must be called before using this function.
  3.1336 +    Value matching(const Edge& edge) const {
  3.1337 +      return Value(edge == (*_matching)[_graph.u(edge)] ? 1 : 0)
  3.1338 +        * primalScale / 2 + Value(edge == (*_matching)[_graph.v(edge)] ? 1 : 0)
  3.1339 +        * primalScale / 2;
  3.1340 +    }
  3.1341 +
  3.1342 +    /// \brief Return the fractional matching arc (or edge) incident
  3.1343 +    /// to the given node.
  3.1344 +    ///
  3.1345 +    /// This function returns one of the fractional matching arc (or
  3.1346 +    /// edge) incident to the given node in the found matching or \c
  3.1347 +    /// INVALID if the node is not covered by the matching or if the
  3.1348 +    /// node is on an odd length cycle then it is the successor edge
  3.1349 +    /// on the cycle.
  3.1350 +    ///
  3.1351 +    /// \pre Either run() or start() must be called before using this function.
  3.1352 +    Arc matching(const Node& node) const {
  3.1353 +      return (*_matching)[node];
  3.1354 +    }
  3.1355 +
  3.1356 +    /// \brief Return a const reference to the matching map.
  3.1357 +    ///
  3.1358 +    /// This function returns a const reference to a node map that stores
  3.1359 +    /// the matching arc (or edge) incident to each node.
  3.1360 +    const MatchingMap& matchingMap() const {
  3.1361 +      return *_matching;
  3.1362 +    }
  3.1363 +
  3.1364 +    /// @}
  3.1365 +
  3.1366 +    /// \name Dual Solution
  3.1367 +    /// Functions to get the dual solution.\n
  3.1368 +    /// Either \ref run() or \ref start() function should be called before
  3.1369 +    /// using them.
  3.1370 +
  3.1371 +    /// @{
  3.1372 +
  3.1373 +    /// \brief Return the value of the dual solution.
  3.1374 +    ///
  3.1375 +    /// This function returns the value of the dual solution.
  3.1376 +    /// It should be equal to the primal value scaled by \ref dualScale
  3.1377 +    /// "dual scale".
  3.1378 +    ///
  3.1379 +    /// \pre Either run() or start() must be called before using this function.
  3.1380 +    Value dualValue() const {
  3.1381 +      Value sum = 0;
  3.1382 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1383 +        sum += nodeValue(n);
  3.1384 +      }
  3.1385 +      return sum;
  3.1386 +    }
  3.1387 +
  3.1388 +    /// \brief Return the dual value (potential) of the given node.
  3.1389 +    ///
  3.1390 +    /// This function returns the dual value (potential) of the given node.
  3.1391 +    ///
  3.1392 +    /// \pre Either run() or start() must be called before using this function.
  3.1393 +    Value nodeValue(const Node& n) const {
  3.1394 +      return (*_node_potential)[n];
  3.1395 +    }
  3.1396 +
  3.1397 +    /// @}
  3.1398 +
  3.1399 +  };
  3.1400 +
  3.1401 +  /// \ingroup matching
  3.1402 +  ///
  3.1403 +  /// \brief Weighted fractional perfect matching in general graphs
  3.1404 +  ///
  3.1405 +  /// This class provides an efficient implementation of fractional
  3.1406 +  /// matching algorithm. The implementation is based on extensive use
  3.1407 +  /// of priority queues and provides \f$O(nm\log n)\f$ time
  3.1408 +  /// complexity.
  3.1409 +  ///
  3.1410 +  /// The maximum weighted fractional perfect matching is a relaxation
  3.1411 +  /// of the maximum weighted perfect matching problem where the odd
  3.1412 +  /// set constraints are omitted.
  3.1413 +  /// It can be formulated with the following linear program.
  3.1414 +  /// \f[ \sum_{e \in \delta(u)}x_e = 1 \quad \forall u\in V\f]
  3.1415 +  /// \f[x_e \ge 0\quad \forall e\in E\f]
  3.1416 +  /// \f[\max \sum_{e\in E}x_ew_e\f]
  3.1417 +  /// where \f$\delta(X)\f$ is the set of edges incident to a node in
  3.1418 +  /// \f$X\f$. The result must be the union of a matching with one
  3.1419 +  /// value edges and a set of odd length cycles with half value edges.
  3.1420 +  ///
  3.1421 +  /// The algorithm calculates an optimal fractional matching and a
  3.1422 +  /// proof of the optimality. The solution of the dual problem can be
  3.1423 +  /// used to check the result of the algorithm. The dual linear
  3.1424 +  /// problem is the following.
  3.1425 +  /// \f[ y_u + y_v \ge w_{uv} \quad \forall uv\in E\f]
  3.1426 +  /// \f[\min \sum_{u \in V}y_u \f] ///
  3.1427 +  ///
  3.1428 +  /// The algorithm can be executed with the run() function.
  3.1429 +  /// After it the matching (the primal solution) and the dual solution
  3.1430 +  /// can be obtained using the query functions.
  3.1431 +
  3.1432 +  /// If the value type is integer, then the primal and the dual
  3.1433 +  /// solutions are multiplied by
  3.1434 +  /// \ref MaxWeightedMatching::primalScale "2" and
  3.1435 +  /// \ref MaxWeightedMatching::dualScale "4" respectively.
  3.1436 +  ///
  3.1437 +  /// \tparam GR The undirected graph type the algorithm runs on.
  3.1438 +  /// \tparam WM The type edge weight map. The default type is
  3.1439 +  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".
  3.1440 +#ifdef DOXYGEN
  3.1441 +  template <typename GR, typename WM>
  3.1442 +#else
  3.1443 +  template <typename GR,
  3.1444 +            typename WM = typename GR::template EdgeMap<int> >
  3.1445 +#endif
  3.1446 +  class MaxWeightedPerfectFractionalMatching {
  3.1447 +  public:
  3.1448 +
  3.1449 +    /// The graph type of the algorithm
  3.1450 +    typedef GR Graph;
  3.1451 +    /// The type of the edge weight map
  3.1452 +    typedef WM WeightMap;
  3.1453 +    /// The value type of the edge weights
  3.1454 +    typedef typename WeightMap::Value Value;
  3.1455 +
  3.1456 +    /// The type of the matching map
  3.1457 +    typedef typename Graph::template NodeMap<typename Graph::Arc>
  3.1458 +    MatchingMap;
  3.1459 +
  3.1460 +    /// \brief Scaling factor for primal solution
  3.1461 +    ///
  3.1462 +    /// Scaling factor for primal solution. It is equal to 2 or 1
  3.1463 +    /// according to the value type.
  3.1464 +    static const int primalScale =
  3.1465 +      std::numeric_limits<Value>::is_integer ? 2 : 1;
  3.1466 +
  3.1467 +    /// \brief Scaling factor for dual solution
  3.1468 +    ///
  3.1469 +    /// Scaling factor for dual solution. It is equal to 4 or 1
  3.1470 +    /// according to the value type.
  3.1471 +    static const int dualScale =
  3.1472 +      std::numeric_limits<Value>::is_integer ? 4 : 1;
  3.1473 +
  3.1474 +  private:
  3.1475 +
  3.1476 +    TEMPLATE_GRAPH_TYPEDEFS(Graph);
  3.1477 +
  3.1478 +    typedef typename Graph::template NodeMap<Value> NodePotential;
  3.1479 +
  3.1480 +    const Graph& _graph;
  3.1481 +    const WeightMap& _weight;
  3.1482 +
  3.1483 +    MatchingMap* _matching;
  3.1484 +    NodePotential* _node_potential;
  3.1485 +
  3.1486 +    int _node_num;
  3.1487 +    bool _allow_loops;
  3.1488 +
  3.1489 +    enum Status {
  3.1490 +      EVEN = -1, MATCHED = 0, ODD = 1
  3.1491 +    };
  3.1492 +
  3.1493 +    typedef typename Graph::template NodeMap<Status> StatusMap;
  3.1494 +    StatusMap* _status;
  3.1495 +
  3.1496 +    typedef typename Graph::template NodeMap<Arc> PredMap;
  3.1497 +    PredMap* _pred;
  3.1498 +
  3.1499 +    typedef ExtendFindEnum<IntNodeMap> TreeSet;
  3.1500 +
  3.1501 +    IntNodeMap *_tree_set_index;
  3.1502 +    TreeSet *_tree_set;
  3.1503 +
  3.1504 +    IntNodeMap *_delta2_index;
  3.1505 +    BinHeap<Value, IntNodeMap> *_delta2;
  3.1506 +
  3.1507 +    IntEdgeMap *_delta3_index;
  3.1508 +    BinHeap<Value, IntEdgeMap> *_delta3;
  3.1509 +
  3.1510 +    Value _delta_sum;
  3.1511 +
  3.1512 +    void createStructures() {
  3.1513 +      _node_num = countNodes(_graph);
  3.1514 +
  3.1515 +      if (!_matching) {
  3.1516 +        _matching = new MatchingMap(_graph);
  3.1517 +      }
  3.1518 +      if (!_node_potential) {
  3.1519 +        _node_potential = new NodePotential(_graph);
  3.1520 +      }
  3.1521 +      if (!_status) {
  3.1522 +        _status = new StatusMap(_graph);
  3.1523 +      }
  3.1524 +      if (!_pred) {
  3.1525 +        _pred = new PredMap(_graph);
  3.1526 +      }
  3.1527 +      if (!_tree_set) {
  3.1528 +        _tree_set_index = new IntNodeMap(_graph);
  3.1529 +        _tree_set = new TreeSet(*_tree_set_index);
  3.1530 +      }
  3.1531 +      if (!_delta2) {
  3.1532 +        _delta2_index = new IntNodeMap(_graph);
  3.1533 +        _delta2 = new BinHeap<Value, IntNodeMap>(*_delta2_index);
  3.1534 +      }
  3.1535 +      if (!_delta3) {
  3.1536 +        _delta3_index = new IntEdgeMap(_graph);
  3.1537 +        _delta3 = new BinHeap<Value, IntEdgeMap>(*_delta3_index);
  3.1538 +      }
  3.1539 +    }
  3.1540 +
  3.1541 +    void destroyStructures() {
  3.1542 +      if (_matching) {
  3.1543 +        delete _matching;
  3.1544 +      }
  3.1545 +      if (_node_potential) {
  3.1546 +        delete _node_potential;
  3.1547 +      }
  3.1548 +      if (_status) {
  3.1549 +        delete _status;
  3.1550 +      }
  3.1551 +      if (_pred) {
  3.1552 +        delete _pred;
  3.1553 +      }
  3.1554 +      if (_tree_set) {
  3.1555 +        delete _tree_set_index;
  3.1556 +        delete _tree_set;
  3.1557 +      }
  3.1558 +      if (_delta2) {
  3.1559 +        delete _delta2_index;
  3.1560 +        delete _delta2;
  3.1561 +      }
  3.1562 +      if (_delta3) {
  3.1563 +        delete _delta3_index;
  3.1564 +        delete _delta3;
  3.1565 +      }
  3.1566 +    }
  3.1567 +
  3.1568 +    void matchedToEven(Node node, int tree) {
  3.1569 +      _tree_set->insert(node, tree);
  3.1570 +      _node_potential->set(node, (*_node_potential)[node] + _delta_sum);
  3.1571 +
  3.1572 +      if (_delta2->state(node) == _delta2->IN_HEAP) {
  3.1573 +        _delta2->erase(node);
  3.1574 +      }
  3.1575 +
  3.1576 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
  3.1577 +        Node v = _graph.source(a);
  3.1578 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
  3.1579 +          dualScale * _weight[a];
  3.1580 +        if (node == v) {
  3.1581 +          if (_allow_loops && _graph.direction(a)) {
  3.1582 +            _delta3->push(a, rw / 2);
  3.1583 +          }
  3.1584 +        } else if ((*_status)[v] == EVEN) {
  3.1585 +          _delta3->push(a, rw / 2);
  3.1586 +        } else if ((*_status)[v] == MATCHED) {
  3.1587 +          if (_delta2->state(v) != _delta2->IN_HEAP) {
  3.1588 +            _pred->set(v, a);
  3.1589 +            _delta2->push(v, rw);
  3.1590 +          } else if ((*_delta2)[v] > rw) {
  3.1591 +            _pred->set(v, a);
  3.1592 +            _delta2->decrease(v, rw);
  3.1593 +          }
  3.1594 +        }
  3.1595 +      }
  3.1596 +    }
  3.1597 +
  3.1598 +    void matchedToOdd(Node node, int tree) {
  3.1599 +      _tree_set->insert(node, tree);
  3.1600 +      _node_potential->set(node, (*_node_potential)[node] - _delta_sum);
  3.1601 +
  3.1602 +      if (_delta2->state(node) == _delta2->IN_HEAP) {
  3.1603 +        _delta2->erase(node);
  3.1604 +      }
  3.1605 +    }
  3.1606 +
  3.1607 +    void evenToMatched(Node node, int tree) {
  3.1608 +      _node_potential->set(node, (*_node_potential)[node] - _delta_sum);
  3.1609 +      Arc min = INVALID;
  3.1610 +      Value minrw = std::numeric_limits<Value>::max();
  3.1611 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
  3.1612 +        Node v = _graph.source(a);
  3.1613 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
  3.1614 +          dualScale * _weight[a];
  3.1615 +
  3.1616 +        if (node == v) {
  3.1617 +          if (_allow_loops && _graph.direction(a)) {
  3.1618 +            _delta3->erase(a);
  3.1619 +          }
  3.1620 +        } else if ((*_status)[v] == EVEN) {
  3.1621 +          _delta3->erase(a);
  3.1622 +          if (minrw > rw) {
  3.1623 +            min = _graph.oppositeArc(a);
  3.1624 +            minrw = rw;
  3.1625 +          }
  3.1626 +        } else if ((*_status)[v]  == MATCHED) {
  3.1627 +          if ((*_pred)[v] == a) {
  3.1628 +            Arc mina = INVALID;
  3.1629 +            Value minrwa = std::numeric_limits<Value>::max();
  3.1630 +            for (OutArcIt aa(_graph, v); aa != INVALID; ++aa) {
  3.1631 +              Node va = _graph.target(aa);
  3.1632 +              if ((*_status)[va] != EVEN ||
  3.1633 +                  _tree_set->find(va) == tree) continue;
  3.1634 +              Value rwa = (*_node_potential)[v] + (*_node_potential)[va] -
  3.1635 +                dualScale * _weight[aa];
  3.1636 +              if (minrwa > rwa) {
  3.1637 +                minrwa = rwa;
  3.1638 +                mina = aa;
  3.1639 +              }
  3.1640 +            }
  3.1641 +            if (mina != INVALID) {
  3.1642 +              _pred->set(v, mina);
  3.1643 +              _delta2->increase(v, minrwa);
  3.1644 +            } else {
  3.1645 +              _pred->set(v, INVALID);
  3.1646 +              _delta2->erase(v);
  3.1647 +            }
  3.1648 +          }
  3.1649 +        }
  3.1650 +      }
  3.1651 +      if (min != INVALID) {
  3.1652 +        _pred->set(node, min);
  3.1653 +        _delta2->push(node, minrw);
  3.1654 +      } else {
  3.1655 +        _pred->set(node, INVALID);
  3.1656 +      }
  3.1657 +    }
  3.1658 +
  3.1659 +    void oddToMatched(Node node) {
  3.1660 +      _node_potential->set(node, (*_node_potential)[node] + _delta_sum);
  3.1661 +      Arc min = INVALID;
  3.1662 +      Value minrw = std::numeric_limits<Value>::max();
  3.1663 +      for (InArcIt a(_graph, node); a != INVALID; ++a) {
  3.1664 +        Node v = _graph.source(a);
  3.1665 +        if ((*_status)[v] != EVEN) continue;
  3.1666 +        Value rw = (*_node_potential)[node] + (*_node_potential)[v] -
  3.1667 +          dualScale * _weight[a];
  3.1668 +
  3.1669 +        if (minrw > rw) {
  3.1670 +          min = _graph.oppositeArc(a);
  3.1671 +          minrw = rw;
  3.1672 +        }
  3.1673 +      }
  3.1674 +      if (min != INVALID) {
  3.1675 +        _pred->set(node, min);
  3.1676 +        _delta2->push(node, minrw);
  3.1677 +      } else {
  3.1678 +        _pred->set(node, INVALID);
  3.1679 +      }
  3.1680 +    }
  3.1681 +
  3.1682 +    void alternatePath(Node even, int tree) {
  3.1683 +      Node odd;
  3.1684 +
  3.1685 +      _status->set(even, MATCHED);
  3.1686 +      evenToMatched(even, tree);
  3.1687 +
  3.1688 +      Arc prev = (*_matching)[even];
  3.1689 +      while (prev != INVALID) {
  3.1690 +        odd = _graph.target(prev);
  3.1691 +        even = _graph.target((*_pred)[odd]);
  3.1692 +        _matching->set(odd, (*_pred)[odd]);
  3.1693 +        _status->set(odd, MATCHED);
  3.1694 +        oddToMatched(odd);
  3.1695 +
  3.1696 +        prev = (*_matching)[even];
  3.1697 +        _status->set(even, MATCHED);
  3.1698 +        _matching->set(even, _graph.oppositeArc((*_matching)[odd]));
  3.1699 +        evenToMatched(even, tree);
  3.1700 +      }
  3.1701 +    }
  3.1702 +
  3.1703 +    void destroyTree(int tree) {
  3.1704 +      for (typename TreeSet::ItemIt n(*_tree_set, tree); n != INVALID; ++n) {
  3.1705 +        if ((*_status)[n] == EVEN) {
  3.1706 +          _status->set(n, MATCHED);
  3.1707 +          evenToMatched(n, tree);
  3.1708 +        } else if ((*_status)[n] == ODD) {
  3.1709 +          _status->set(n, MATCHED);
  3.1710 +          oddToMatched(n);
  3.1711 +        }
  3.1712 +      }
  3.1713 +      _tree_set->eraseClass(tree);
  3.1714 +    }
  3.1715 +
  3.1716 +    void augmentOnEdge(const Edge& edge) {
  3.1717 +      Node left = _graph.u(edge);
  3.1718 +      int left_tree = _tree_set->find(left);
  3.1719 +
  3.1720 +      alternatePath(left, left_tree);
  3.1721 +      destroyTree(left_tree);
  3.1722 +      _matching->set(left, _graph.direct(edge, true));
  3.1723 +
  3.1724 +      Node right = _graph.v(edge);
  3.1725 +      int right_tree = _tree_set->find(right);
  3.1726 +
  3.1727 +      alternatePath(right, right_tree);
  3.1728 +      destroyTree(right_tree);
  3.1729 +      _matching->set(right, _graph.direct(edge, false));
  3.1730 +    }
  3.1731 +
  3.1732 +    void augmentOnArc(const Arc& arc) {
  3.1733 +      Node left = _graph.source(arc);
  3.1734 +      _status->set(left, MATCHED);
  3.1735 +      _matching->set(left, arc);
  3.1736 +      _pred->set(left, arc);
  3.1737 +
  3.1738 +      Node right = _graph.target(arc);
  3.1739 +      int right_tree = _tree_set->find(right);
  3.1740 +
  3.1741 +      alternatePath(right, right_tree);
  3.1742 +      destroyTree(right_tree);
  3.1743 +      _matching->set(right, _graph.oppositeArc(arc));
  3.1744 +    }
  3.1745 +
  3.1746 +    void extendOnArc(const Arc& arc) {
  3.1747 +      Node base = _graph.target(arc);
  3.1748 +      int tree = _tree_set->find(base);
  3.1749 +
  3.1750 +      Node odd = _graph.source(arc);
  3.1751 +      _tree_set->insert(odd, tree);
  3.1752 +      _status->set(odd, ODD);
  3.1753 +      matchedToOdd(odd, tree);
  3.1754 +      _pred->set(odd, arc);
  3.1755 +
  3.1756 +      Node even = _graph.target((*_matching)[odd]);
  3.1757 +      _tree_set->insert(even, tree);
  3.1758 +      _status->set(even, EVEN);
  3.1759 +      matchedToEven(even, tree);
  3.1760 +    }
  3.1761 +
  3.1762 +    void cycleOnEdge(const Edge& edge, int tree) {
  3.1763 +      Node nca = INVALID;
  3.1764 +      std::vector<Node> left_path, right_path;
  3.1765 +
  3.1766 +      {
  3.1767 +        std::set<Node> left_set, right_set;
  3.1768 +        Node left = _graph.u(edge);
  3.1769 +        left_path.push_back(left);
  3.1770 +        left_set.insert(left);
  3.1771 +
  3.1772 +        Node right = _graph.v(edge);
  3.1773 +        right_path.push_back(right);
  3.1774 +        right_set.insert(right);
  3.1775 +
  3.1776 +        while (true) {
  3.1777 +
  3.1778 +          if (left_set.find(right) != left_set.end()) {
  3.1779 +            nca = right;
  3.1780 +            break;
  3.1781 +          }
  3.1782 +
  3.1783 +          if ((*_matching)[left] == INVALID) break;
  3.1784 +
  3.1785 +          left = _graph.target((*_matching)[left]);
  3.1786 +          left_path.push_back(left);
  3.1787 +          left = _graph.target((*_pred)[left]);
  3.1788 +          left_path.push_back(left);
  3.1789 +
  3.1790 +          left_set.insert(left);
  3.1791 +
  3.1792 +          if (right_set.find(left) != right_set.end()) {
  3.1793 +            nca = left;
  3.1794 +            break;
  3.1795 +          }
  3.1796 +
  3.1797 +          if ((*_matching)[right] == INVALID) break;
  3.1798 +
  3.1799 +          right = _graph.target((*_matching)[right]);
  3.1800 +          right_path.push_back(right);
  3.1801 +          right = _graph.target((*_pred)[right]);
  3.1802 +          right_path.push_back(right);
  3.1803 +
  3.1804 +          right_set.insert(right);
  3.1805 +
  3.1806 +        }
  3.1807 +
  3.1808 +        if (nca == INVALID) {
  3.1809 +          if ((*_matching)[left] == INVALID) {
  3.1810 +            nca = right;
  3.1811 +            while (left_set.find(nca) == left_set.end()) {
  3.1812 +              nca = _graph.target((*_matching)[nca]);
  3.1813 +              right_path.push_back(nca);
  3.1814 +              nca = _graph.target((*_pred)[nca]);
  3.1815 +              right_path.push_back(nca);
  3.1816 +            }
  3.1817 +          } else {
  3.1818 +            nca = left;
  3.1819 +            while (right_set.find(nca) == right_set.end()) {
  3.1820 +              nca = _graph.target((*_matching)[nca]);
  3.1821 +              left_path.push_back(nca);
  3.1822 +              nca = _graph.target((*_pred)[nca]);
  3.1823 +              left_path.push_back(nca);
  3.1824 +            }
  3.1825 +          }
  3.1826 +        }
  3.1827 +      }
  3.1828 +
  3.1829 +      alternatePath(nca, tree);
  3.1830 +      Arc prev;
  3.1831 +
  3.1832 +      prev = _graph.direct(edge, true);
  3.1833 +      for (int i = 0; left_path[i] != nca; i += 2) {
  3.1834 +        _matching->set(left_path[i], prev);
  3.1835 +        _status->set(left_path[i], MATCHED);
  3.1836 +        evenToMatched(left_path[i], tree);
  3.1837 +
  3.1838 +        prev = _graph.oppositeArc((*_pred)[left_path[i + 1]]);
  3.1839 +        _status->set(left_path[i + 1], MATCHED);
  3.1840 +        oddToMatched(left_path[i + 1]);
  3.1841 +      }
  3.1842 +      _matching->set(nca, prev);
  3.1843 +
  3.1844 +      for (int i = 0; right_path[i] != nca; i += 2) {
  3.1845 +        _status->set(right_path[i], MATCHED);
  3.1846 +        evenToMatched(right_path[i], tree);
  3.1847 +
  3.1848 +        _matching->set(right_path[i + 1], (*_pred)[right_path[i + 1]]);
  3.1849 +        _status->set(right_path[i + 1], MATCHED);
  3.1850 +        oddToMatched(right_path[i + 1]);
  3.1851 +      }
  3.1852 +
  3.1853 +      destroyTree(tree);
  3.1854 +    }
  3.1855 +
  3.1856 +    void extractCycle(const Arc &arc) {
  3.1857 +      Node left = _graph.source(arc);
  3.1858 +      Node odd = _graph.target((*_matching)[left]);
  3.1859 +      Arc prev;
  3.1860 +      while (odd != left) {
  3.1861 +        Node even = _graph.target((*_matching)[odd]);
  3.1862 +        prev = (*_matching)[odd];
  3.1863 +        odd = _graph.target((*_matching)[even]);
  3.1864 +        _matching->set(even, _graph.oppositeArc(prev));
  3.1865 +      }
  3.1866 +      _matching->set(left, arc);
  3.1867 +
  3.1868 +      Node right = _graph.target(arc);
  3.1869 +      int right_tree = _tree_set->find(right);
  3.1870 +      alternatePath(right, right_tree);
  3.1871 +      destroyTree(right_tree);
  3.1872 +      _matching->set(right, _graph.oppositeArc(arc));
  3.1873 +    }
  3.1874 +
  3.1875 +  public:
  3.1876 +
  3.1877 +    /// \brief Constructor
  3.1878 +    ///
  3.1879 +    /// Constructor.
  3.1880 +    MaxWeightedPerfectFractionalMatching(const Graph& graph,
  3.1881 +                                         const WeightMap& weight,
  3.1882 +                                         bool allow_loops = true)
  3.1883 +      : _graph(graph), _weight(weight), _matching(0),
  3.1884 +      _node_potential(0), _node_num(0), _allow_loops(allow_loops),
  3.1885 +      _status(0),  _pred(0),
  3.1886 +      _tree_set_index(0), _tree_set(0),
  3.1887 +
  3.1888 +      _delta2_index(0), _delta2(0),
  3.1889 +      _delta3_index(0), _delta3(0),
  3.1890 +
  3.1891 +      _delta_sum() {}
  3.1892 +
  3.1893 +    ~MaxWeightedPerfectFractionalMatching() {
  3.1894 +      destroyStructures();
  3.1895 +    }
  3.1896 +
  3.1897 +    /// \name Execution Control
  3.1898 +    /// The simplest way to execute the algorithm is to use the
  3.1899 +    /// \ref run() member function.
  3.1900 +
  3.1901 +    ///@{
  3.1902 +
  3.1903 +    /// \brief Initialize the algorithm
  3.1904 +    ///
  3.1905 +    /// This function initializes the algorithm.
  3.1906 +    void init() {
  3.1907 +      createStructures();
  3.1908 +
  3.1909 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1910 +        (*_delta2_index)[n] = _delta2->PRE_HEAP;
  3.1911 +      }
  3.1912 +      for (EdgeIt e(_graph); e != INVALID; ++e) {
  3.1913 +        (*_delta3_index)[e] = _delta3->PRE_HEAP;
  3.1914 +      }
  3.1915 +
  3.1916 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.1917 +        Value max = - std::numeric_limits<Value>::max();
  3.1918 +        for (OutArcIt e(_graph, n); e != INVALID; ++e) {
  3.1919 +          if (_graph.target(e) == n && !_allow_loops) continue;
  3.1920 +          if ((dualScale * _weight[e]) / 2 > max) {
  3.1921 +            max = (dualScale * _weight[e]) / 2;
  3.1922 +          }
  3.1923 +        }
  3.1924 +        _node_potential->set(n, max);
  3.1925 +
  3.1926 +        _tree_set->insert(n);
  3.1927 +
  3.1928 +        _matching->set(n, INVALID);
  3.1929 +        _status->set(n, EVEN);
  3.1930 +      }
  3.1931 +
  3.1932 +      for (EdgeIt e(_graph); e != INVALID; ++e) {
  3.1933 +        Node left = _graph.u(e);
  3.1934 +        Node right = _graph.v(e);
  3.1935 +        if (left == right && !_allow_loops) continue;
  3.1936 +        _delta3->push(e, ((*_node_potential)[left] +
  3.1937 +                          (*_node_potential)[right] -
  3.1938 +                          dualScale * _weight[e]) / 2);
  3.1939 +      }
  3.1940 +    }
  3.1941 +
  3.1942 +    /// \brief Start the algorithm
  3.1943 +    ///
  3.1944 +    /// This function starts the algorithm.
  3.1945 +    ///
  3.1946 +    /// \pre \ref init() must be called before using this function.
  3.1947 +    bool start() {
  3.1948 +      enum OpType {
  3.1949 +        D2, D3
  3.1950 +      };
  3.1951 +
  3.1952 +      int unmatched = _node_num;
  3.1953 +      while (unmatched > 0) {
  3.1954 +        Value d2 = !_delta2->empty() ?
  3.1955 +          _delta2->prio() : std::numeric_limits<Value>::max();
  3.1956 +
  3.1957 +        Value d3 = !_delta3->empty() ?
  3.1958 +          _delta3->prio() : std::numeric_limits<Value>::max();
  3.1959 +
  3.1960 +        _delta_sum = d3; OpType ot = D3;
  3.1961 +        if (d2 < _delta_sum) { _delta_sum = d2; ot = D2; }
  3.1962 +
  3.1963 +        if (_delta_sum == std::numeric_limits<Value>::max()) {
  3.1964 +          return false;
  3.1965 +        }
  3.1966 +
  3.1967 +        switch (ot) {
  3.1968 +        case D2:
  3.1969 +          {
  3.1970 +            Node n = _delta2->top();
  3.1971 +            Arc a = (*_pred)[n];
  3.1972 +            if ((*_matching)[n] == INVALID) {
  3.1973 +              augmentOnArc(a);
  3.1974 +              --unmatched;
  3.1975 +            } else {
  3.1976 +              Node v = _graph.target((*_matching)[n]);
  3.1977 +              if ((*_matching)[n] !=
  3.1978 +                  _graph.oppositeArc((*_matching)[v])) {
  3.1979 +                extractCycle(a);
  3.1980 +                --unmatched;
  3.1981 +              } else {
  3.1982 +                extendOnArc(a);
  3.1983 +              }
  3.1984 +            }
  3.1985 +          } break;
  3.1986 +        case D3:
  3.1987 +          {
  3.1988 +            Edge e = _delta3->top();
  3.1989 +
  3.1990 +            Node left = _graph.u(e);
  3.1991 +            Node right = _graph.v(e);
  3.1992 +
  3.1993 +            int left_tree = _tree_set->find(left);
  3.1994 +            int right_tree = _tree_set->find(right);
  3.1995 +
  3.1996 +            if (left_tree == right_tree) {
  3.1997 +              cycleOnEdge(e, left_tree);
  3.1998 +              --unmatched;
  3.1999 +            } else {
  3.2000 +              augmentOnEdge(e);
  3.2001 +              unmatched -= 2;
  3.2002 +            }
  3.2003 +          } break;
  3.2004 +        }
  3.2005 +      }
  3.2006 +      return true;
  3.2007 +    }
  3.2008 +
  3.2009 +    /// \brief Run the algorithm.
  3.2010 +    ///
  3.2011 +    /// This method runs the \c %MaxWeightedMatching algorithm.
  3.2012 +    ///
  3.2013 +    /// \note mwfm.run() is just a shortcut of the following code.
  3.2014 +    /// \code
  3.2015 +    ///   mwpfm.init();
  3.2016 +    ///   mwpfm.start();
  3.2017 +    /// \endcode
  3.2018 +    bool run() {
  3.2019 +      init();
  3.2020 +      return start();
  3.2021 +    }
  3.2022 +
  3.2023 +    /// @}
  3.2024 +
  3.2025 +    /// \name Primal Solution
  3.2026 +    /// Functions to get the primal solution, i.e. the maximum weighted
  3.2027 +    /// matching.\n
  3.2028 +    /// Either \ref run() or \ref start() function should be called before
  3.2029 +    /// using them.
  3.2030 +
  3.2031 +    /// @{
  3.2032 +
  3.2033 +    /// \brief Return the weight of the matching.
  3.2034 +    ///
  3.2035 +    /// This function returns the weight of the found matching. This
  3.2036 +    /// value is scaled by \ref primalScale "primal scale".
  3.2037 +    ///
  3.2038 +    /// \pre Either run() or start() must be called before using this function.
  3.2039 +    Value matchingWeight() const {
  3.2040 +      Value sum = 0;
  3.2041 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.2042 +        if ((*_matching)[n] != INVALID) {
  3.2043 +          sum += _weight[(*_matching)[n]];
  3.2044 +        }
  3.2045 +      }
  3.2046 +      return sum * primalScale / 2;
  3.2047 +    }
  3.2048 +
  3.2049 +    /// \brief Return the number of covered nodes in the matching.
  3.2050 +    ///
  3.2051 +    /// This function returns the number of covered nodes in the matching.
  3.2052 +    ///
  3.2053 +    /// \pre Either run() or start() must be called before using this function.
  3.2054 +    int matchingSize() const {
  3.2055 +      int num = 0;
  3.2056 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.2057 +        if ((*_matching)[n] != INVALID) {
  3.2058 +          ++num;
  3.2059 +        }
  3.2060 +      }
  3.2061 +      return num;
  3.2062 +    }
  3.2063 +
  3.2064 +    /// \brief Return \c true if the given edge is in the matching.
  3.2065 +    ///
  3.2066 +    /// This function returns \c true if the given edge is in the
  3.2067 +    /// found matching. The result is scaled by \ref primalScale
  3.2068 +    /// "primal scale".
  3.2069 +    ///
  3.2070 +    /// \pre Either run() or start() must be called before using this function.
  3.2071 +    Value matching(const Edge& edge) const {
  3.2072 +      return Value(edge == (*_matching)[_graph.u(edge)] ? 1 : 0)
  3.2073 +        * primalScale / 2 + Value(edge == (*_matching)[_graph.v(edge)] ? 1 : 0)
  3.2074 +        * primalScale / 2;
  3.2075 +    }
  3.2076 +
  3.2077 +    /// \brief Return the fractional matching arc (or edge) incident
  3.2078 +    /// to the given node.
  3.2079 +    ///
  3.2080 +    /// This function returns one of the fractional matching arc (or
  3.2081 +    /// edge) incident to the given node in the found matching or \c
  3.2082 +    /// INVALID if the node is not covered by the matching or if the
  3.2083 +    /// node is on an odd length cycle then it is the successor edge
  3.2084 +    /// on the cycle.
  3.2085 +    ///
  3.2086 +    /// \pre Either run() or start() must be called before using this function.
  3.2087 +    Arc matching(const Node& node) const {
  3.2088 +      return (*_matching)[node];
  3.2089 +    }
  3.2090 +
  3.2091 +    /// \brief Return a const reference to the matching map.
  3.2092 +    ///
  3.2093 +    /// This function returns a const reference to a node map that stores
  3.2094 +    /// the matching arc (or edge) incident to each node.
  3.2095 +    const MatchingMap& matchingMap() const {
  3.2096 +      return *_matching;
  3.2097 +    }
  3.2098 +
  3.2099 +    /// @}
  3.2100 +
  3.2101 +    /// \name Dual Solution
  3.2102 +    /// Functions to get the dual solution.\n
  3.2103 +    /// Either \ref run() or \ref start() function should be called before
  3.2104 +    /// using them.
  3.2105 +
  3.2106 +    /// @{
  3.2107 +
  3.2108 +    /// \brief Return the value of the dual solution.
  3.2109 +    ///
  3.2110 +    /// This function returns the value of the dual solution.
  3.2111 +    /// It should be equal to the primal value scaled by \ref dualScale
  3.2112 +    /// "dual scale".
  3.2113 +    ///
  3.2114 +    /// \pre Either run() or start() must be called before using this function.
  3.2115 +    Value dualValue() const {
  3.2116 +      Value sum = 0;
  3.2117 +      for (NodeIt n(_graph); n != INVALID; ++n) {
  3.2118 +        sum += nodeValue(n);
  3.2119 +      }
  3.2120 +      return sum;
  3.2121 +    }
  3.2122 +
  3.2123 +    /// \brief Return the dual value (potential) of the given node.
  3.2124 +    ///
  3.2125 +    /// This function returns the dual value (potential) of the given node.
  3.2126 +    ///
  3.2127 +    /// \pre Either run() or start() must be called before using this function.
  3.2128 +    Value nodeValue(const Node& n) const {
  3.2129 +      return (*_node_potential)[n];
  3.2130 +    }
  3.2131 +
  3.2132 +    /// @}
  3.2133 +
  3.2134 +  };
  3.2135 +
  3.2136 +} //END OF NAMESPACE LEMON
  3.2137 +
  3.2138 +#endif //LEMON_FRACTIONAL_MATCHING_H
     4.1 --- a/test/CMakeLists.txt	Sun Sep 20 21:38:24 2009 +0200
     4.2 +++ b/test/CMakeLists.txt	Fri Sep 25 21:51:36 2009 +0200
     4.3 @@ -21,6 +21,7 @@
     4.4    edge_set_test
     4.5    error_test
     4.6    euler_test
     4.7 +  fractional_matching_test
     4.8    gomory_hu_test
     4.9    graph_copy_test
    4.10    graph_test
     5.1 --- a/test/Makefile.am	Sun Sep 20 21:38:24 2009 +0200
     5.2 +++ b/test/Makefile.am	Fri Sep 25 21:51:36 2009 +0200
     5.3 @@ -19,6 +19,7 @@
     5.4  	test/edge_set_test \
     5.5  	test/error_test \
     5.6  	test/euler_test \
     5.7 +	test/fractional_matching_test \
     5.8  	test/gomory_hu_test \
     5.9  	test/graph_copy_test \
    5.10  	test/graph_test \
    5.11 @@ -65,6 +66,7 @@
    5.12  test_edge_set_test_SOURCES = test/edge_set_test.cc
    5.13  test_error_test_SOURCES = test/error_test.cc
    5.14  test_euler_test_SOURCES = test/euler_test.cc
    5.15 +test_fractional_matching_test_SOURCES = test/fractional_matching_test.cc
    5.16  test_gomory_hu_test_SOURCES = test/gomory_hu_test.cc
    5.17  test_graph_copy_test_SOURCES = test/graph_copy_test.cc
    5.18  test_graph_test_SOURCES = test/graph_test.cc
     6.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     6.2 +++ b/test/fractional_matching_test.cc	Fri Sep 25 21:51:36 2009 +0200
     6.3 @@ -0,0 +1,502 @@
     6.4 +/* -*- mode: C++; indent-tabs-mode: nil; -*-
     6.5 + *
     6.6 + * This file is a part of LEMON, a generic C++ optimization library.
     6.7 + *
     6.8 + * Copyright (C) 2003-2009
     6.9 + * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    6.10 + * (Egervary Research Group on Combinatorial Optimization, EGRES).
    6.11 + *
    6.12 + * Permission to use, modify and distribute this software is granted
    6.13 + * provided that this copyright notice appears in all copies. For
    6.14 + * precise terms see the accompanying LICENSE file.
    6.15 + *
    6.16 + * This software is provided "AS IS" with no warranty of any kind,
    6.17 + * express or implied, and with no claim as to its suitability for any
    6.18 + * purpose.
    6.19 + *
    6.20 + */
    6.21 +
    6.22 +#include <iostream>
    6.23 +#include <sstream>
    6.24 +#include <vector>
    6.25 +#include <queue>
    6.26 +#include <cstdlib>
    6.27 +
    6.28 +#include <lemon/fractional_matching.h>
    6.29 +#include <lemon/smart_graph.h>
    6.30 +#include <lemon/concepts/graph.h>
    6.31 +#include <lemon/concepts/maps.h>
    6.32 +#include <lemon/lgf_reader.h>
    6.33 +#include <lemon/math.h>
    6.34 +
    6.35 +#include "test_tools.h"
    6.36 +
    6.37 +using namespace std;
    6.38 +using namespace lemon;
    6.39 +
    6.40 +GRAPH_TYPEDEFS(SmartGraph);
    6.41 +
    6.42 +
    6.43 +const int lgfn = 4;
    6.44 +const std::string lgf[lgfn] = {
    6.45 +  "@nodes\n"
    6.46 +  "label\n"
    6.47 +  "0\n"
    6.48 +  "1\n"
    6.49 +  "2\n"
    6.50 +  "3\n"
    6.51 +  "4\n"
    6.52 +  "5\n"
    6.53 +  "6\n"
    6.54 +  "7\n"
    6.55 +  "@edges\n"
    6.56 +  "     label  weight\n"
    6.57 +  "7 4  0      984\n"
    6.58 +  "0 7  1      73\n"
    6.59 +  "7 1  2      204\n"
    6.60 +  "2 3  3      583\n"
    6.61 +  "2 7  4      565\n"
    6.62 +  "2 1  5      582\n"
    6.63 +  "0 4  6      551\n"
    6.64 +  "2 5  7      385\n"
    6.65 +  "1 5  8      561\n"
    6.66 +  "5 3  9      484\n"
    6.67 +  "7 5  10     904\n"
    6.68 +  "3 6  11     47\n"
    6.69 +  "7 6  12     888\n"
    6.70 +  "3 0  13     747\n"
    6.71 +  "6 1  14     310\n",
    6.72 +
    6.73 +  "@nodes\n"
    6.74 +  "label\n"
    6.75 +  "0\n"
    6.76 +  "1\n"
    6.77 +  "2\n"
    6.78 +  "3\n"
    6.79 +  "4\n"
    6.80 +  "5\n"
    6.81 +  "6\n"
    6.82 +  "7\n"
    6.83 +  "@edges\n"
    6.84 +  "     label  weight\n"
    6.85 +  "2 5  0      710\n"
    6.86 +  "0 5  1      241\n"
    6.87 +  "2 4  2      856\n"
    6.88 +  "2 6  3      762\n"
    6.89 +  "4 1  4      747\n"
    6.90 +  "6 1  5      962\n"
    6.91 +  "4 7  6      723\n"
    6.92 +  "1 7  7      661\n"
    6.93 +  "2 3  8      376\n"
    6.94 +  "1 0  9      416\n"
    6.95 +  "6 7  10     391\n",
    6.96 +
    6.97 +  "@nodes\n"
    6.98 +  "label\n"
    6.99 +  "0\n"
   6.100 +  "1\n"
   6.101 +  "2\n"
   6.102 +  "3\n"
   6.103 +  "4\n"
   6.104 +  "5\n"
   6.105 +  "6\n"
   6.106 +  "7\n"
   6.107 +  "@edges\n"
   6.108 +  "     label  weight\n"
   6.109 +  "6 2  0      553\n"
   6.110 +  "0 7  1      653\n"
   6.111 +  "6 3  2      22\n"
   6.112 +  "4 7  3      846\n"
   6.113 +  "7 2  4      981\n"
   6.114 +  "7 6  5      250\n"
   6.115 +  "5 2  6      539\n",
   6.116 +
   6.117 +  "@nodes\n"
   6.118 +  "label\n"
   6.119 +  "0\n"
   6.120 +  "@edges\n"
   6.121 +  "     label  weight\n"
   6.122 +  "0 0  0      100\n"
   6.123 +};
   6.124 +
   6.125 +void checkMaxFractionalMatchingCompile()
   6.126 +{
   6.127 +  typedef concepts::Graph Graph;
   6.128 +  typedef Graph::Node Node;
   6.129 +  typedef Graph::Edge Edge;
   6.130 +
   6.131 +  Graph g;
   6.132 +  Node n;
   6.133 +  Edge e;
   6.134 +
   6.135 +  MaxFractionalMatching<Graph> mat_test(g);
   6.136 +  const MaxFractionalMatching<Graph>&
   6.137 +    const_mat_test = mat_test;
   6.138 +
   6.139 +  mat_test.init();
   6.140 +  mat_test.start();
   6.141 +  mat_test.start(true);
   6.142 +  mat_test.startPerfect();
   6.143 +  mat_test.startPerfect(true);
   6.144 +  mat_test.run();
   6.145 +  mat_test.run(true);
   6.146 +  mat_test.runPerfect();
   6.147 +  mat_test.runPerfect(true);
   6.148 +
   6.149 +  const_mat_test.matchingSize();
   6.150 +  const_mat_test.matching(e);
   6.151 +  const_mat_test.matching(n);
   6.152 +  const MaxFractionalMatching<Graph>::MatchingMap& mmap =
   6.153 +    const_mat_test.matchingMap();
   6.154 +  e = mmap[n];
   6.155 +
   6.156 +  const_mat_test.barrier(n);
   6.157 +}
   6.158 +
   6.159 +void checkMaxWeightedFractionalMatchingCompile()
   6.160 +{
   6.161 +  typedef concepts::Graph Graph;
   6.162 +  typedef Graph::Node Node;
   6.163 +  typedef Graph::Edge Edge;
   6.164 +  typedef Graph::EdgeMap<int> WeightMap;
   6.165 +
   6.166 +  Graph g;
   6.167 +  Node n;
   6.168 +  Edge e;
   6.169 +  WeightMap w(g);
   6.170 +
   6.171 +  MaxWeightedFractionalMatching<Graph> mat_test(g, w);
   6.172 +  const MaxWeightedFractionalMatching<Graph>&
   6.173 +    const_mat_test = mat_test;
   6.174 +
   6.175 +  mat_test.init();
   6.176 +  mat_test.start();
   6.177 +  mat_test.run();
   6.178 +
   6.179 +  const_mat_test.matchingWeight();
   6.180 +  const_mat_test.matchingSize();
   6.181 +  const_mat_test.matching(e);
   6.182 +  const_mat_test.matching(n);
   6.183 +  const MaxWeightedFractionalMatching<Graph>::MatchingMap& mmap =
   6.184 +    const_mat_test.matchingMap();
   6.185 +  e = mmap[n];
   6.186 +
   6.187 +  const_mat_test.dualValue();
   6.188 +  const_mat_test.nodeValue(n);
   6.189 +}
   6.190 +
   6.191 +void checkMaxWeightedPerfectFractionalMatchingCompile()
   6.192 +{
   6.193 +  typedef concepts::Graph Graph;
   6.194 +  typedef Graph::Node Node;
   6.195 +  typedef Graph::Edge Edge;
   6.196 +  typedef Graph::EdgeMap<int> WeightMap;
   6.197 +
   6.198 +  Graph g;
   6.199 +  Node n;
   6.200 +  Edge e;
   6.201 +  WeightMap w(g);
   6.202 +
   6.203 +  MaxWeightedPerfectFractionalMatching<Graph> mat_test(g, w);
   6.204 +  const MaxWeightedPerfectFractionalMatching<Graph>&
   6.205 +    const_mat_test = mat_test;
   6.206 +
   6.207 +  mat_test.init();
   6.208 +  mat_test.start();
   6.209 +  mat_test.run();
   6.210 +
   6.211 +  const_mat_test.matchingWeight();
   6.212 +  const_mat_test.matching(e);
   6.213 +  const_mat_test.matching(n);
   6.214 +  const MaxWeightedPerfectFractionalMatching<Graph>::MatchingMap& mmap =
   6.215 +    const_mat_test.matchingMap();
   6.216 +  e = mmap[n];
   6.217 +
   6.218 +  const_mat_test.dualValue();
   6.219 +  const_mat_test.nodeValue(n);
   6.220 +}
   6.221 +
   6.222 +void checkFractionalMatching(const SmartGraph& graph,
   6.223 +                             const MaxFractionalMatching<SmartGraph>& mfm,
   6.224 +                             bool allow_loops = true) {
   6.225 +  int pv = 0;
   6.226 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.227 +    int indeg = 0;
   6.228 +    for (InArcIt a(graph, n); a != INVALID; ++a) {
   6.229 +      if (mfm.matching(graph.source(a)) == a) {
   6.230 +        ++indeg;
   6.231 +      }
   6.232 +    }
   6.233 +    if (mfm.matching(n) != INVALID) {
   6.234 +      check(indeg == 1, "Invalid matching");
   6.235 +      ++pv;
   6.236 +    } else {
   6.237 +      check(indeg == 0, "Invalid matching");
   6.238 +    }
   6.239 +  }
   6.240 +  check(pv == mfm.matchingSize(), "Wrong matching size");
   6.241 +
   6.242 +  SmartGraph::NodeMap<bool> processed(graph, false);
   6.243 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.244 +    if (processed[n]) continue;
   6.245 +    processed[n] = true;
   6.246 +    if (mfm.matching(n) == INVALID) continue;
   6.247 +    int num = 1;
   6.248 +    Node v = graph.target(mfm.matching(n));
   6.249 +    while (v != n) {
   6.250 +      processed[v] = true;
   6.251 +      ++num;
   6.252 +      v = graph.target(mfm.matching(v));
   6.253 +    }
   6.254 +    check(num == 2 || num % 2 == 1, "Wrong cycle size");
   6.255 +    check(allow_loops || num != 1, "Wrong cycle size");
   6.256 +  }
   6.257 +
   6.258 +  int anum = 0, bnum = 0;
   6.259 +  SmartGraph::NodeMap<bool> neighbours(graph, false);
   6.260 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.261 +    if (!mfm.barrier(n)) continue;
   6.262 +    ++anum;
   6.263 +    for (SmartGraph::InArcIt a(graph, n); a != INVALID; ++a) {
   6.264 +      Node u = graph.source(a);
   6.265 +      if (!allow_loops && u == n) continue;
   6.266 +      if (!neighbours[u]) {
   6.267 +        neighbours[u] = true;
   6.268 +        ++bnum;
   6.269 +      }
   6.270 +    }
   6.271 +  }
   6.272 +  check(anum - bnum + mfm.matchingSize() == countNodes(graph),
   6.273 +        "Wrong barrier");
   6.274 +}
   6.275 +
   6.276 +void checkPerfectFractionalMatching(const SmartGraph& graph,
   6.277 +                             const MaxFractionalMatching<SmartGraph>& mfm,
   6.278 +                             bool perfect, bool allow_loops = true) {
   6.279 +  if (perfect) {
   6.280 +    for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.281 +      int indeg = 0;
   6.282 +      for (InArcIt a(graph, n); a != INVALID; ++a) {
   6.283 +        if (mfm.matching(graph.source(a)) == a) {
   6.284 +          ++indeg;
   6.285 +        }
   6.286 +      }
   6.287 +      check(mfm.matching(n) != INVALID, "Invalid matching");
   6.288 +      check(indeg == 1, "Invalid matching");
   6.289 +    }
   6.290 +  } else {
   6.291 +    int anum = 0, bnum = 0;
   6.292 +    SmartGraph::NodeMap<bool> neighbours(graph, false);
   6.293 +    for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.294 +      if (!mfm.barrier(n)) continue;
   6.295 +      ++anum;
   6.296 +      for (SmartGraph::InArcIt a(graph, n); a != INVALID; ++a) {
   6.297 +        Node u = graph.source(a);
   6.298 +        if (!allow_loops && u == n) continue;
   6.299 +        if (!neighbours[u]) {
   6.300 +          neighbours[u] = true;
   6.301 +          ++bnum;
   6.302 +        }
   6.303 +      }
   6.304 +    }
   6.305 +    check(anum - bnum > 0, "Wrong barrier");
   6.306 +  }
   6.307 +}
   6.308 +
   6.309 +void checkWeightedFractionalMatching(const SmartGraph& graph,
   6.310 +                   const SmartGraph::EdgeMap<int>& weight,
   6.311 +                   const MaxWeightedFractionalMatching<SmartGraph>& mwfm,
   6.312 +                   bool allow_loops = true) {
   6.313 +  for (SmartGraph::EdgeIt e(graph); e != INVALID; ++e) {
   6.314 +    if (graph.u(e) == graph.v(e) && !allow_loops) continue;
   6.315 +    int rw = mwfm.nodeValue(graph.u(e)) + mwfm.nodeValue(graph.v(e))
   6.316 +      - weight[e] * mwfm.dualScale;
   6.317 +
   6.318 +    check(rw >= 0, "Negative reduced weight");
   6.319 +    check(rw == 0 || !mwfm.matching(e),
   6.320 +          "Non-zero reduced weight on matching edge");
   6.321 +  }
   6.322 +
   6.323 +  int pv = 0;
   6.324 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.325 +    int indeg = 0;
   6.326 +    for (InArcIt a(graph, n); a != INVALID; ++a) {
   6.327 +      if (mwfm.matching(graph.source(a)) == a) {
   6.328 +        ++indeg;
   6.329 +      }
   6.330 +    }
   6.331 +    check(indeg <= 1, "Invalid matching");
   6.332 +    if (mwfm.matching(n) != INVALID) {
   6.333 +      check(mwfm.nodeValue(n) >= 0, "Invalid node value");
   6.334 +      check(indeg == 1, "Invalid matching");
   6.335 +      pv += weight[mwfm.matching(n)];
   6.336 +      SmartGraph::Node o = graph.target(mwfm.matching(n));
   6.337 +    } else {
   6.338 +      check(mwfm.nodeValue(n) == 0, "Invalid matching");
   6.339 +      check(indeg == 0, "Invalid matching");
   6.340 +    }
   6.341 +  }
   6.342 +
   6.343 +  int dv = 0;
   6.344 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.345 +    dv += mwfm.nodeValue(n);
   6.346 +  }
   6.347 +
   6.348 +  check(pv * mwfm.dualScale == dv * 2, "Wrong duality");
   6.349 +
   6.350 +  SmartGraph::NodeMap<bool> processed(graph, false);
   6.351 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.352 +    if (processed[n]) continue;
   6.353 +    processed[n] = true;
   6.354 +    if (mwfm.matching(n) == INVALID) continue;
   6.355 +    int num = 1;
   6.356 +    Node v = graph.target(mwfm.matching(n));
   6.357 +    while (v != n) {
   6.358 +      processed[v] = true;
   6.359 +      ++num;
   6.360 +      v = graph.target(mwfm.matching(v));
   6.361 +    }
   6.362 +    check(num == 2 || num % 2 == 1, "Wrong cycle size");
   6.363 +    check(allow_loops || num != 1, "Wrong cycle size");
   6.364 +  }
   6.365 +
   6.366 +  return;
   6.367 +}
   6.368 +
   6.369 +void checkWeightedPerfectFractionalMatching(const SmartGraph& graph,
   6.370 +                const SmartGraph::EdgeMap<int>& weight,
   6.371 +                const MaxWeightedPerfectFractionalMatching<SmartGraph>& mwpfm,
   6.372 +                bool allow_loops = true) {
   6.373 +  for (SmartGraph::EdgeIt e(graph); e != INVALID; ++e) {
   6.374 +    if (graph.u(e) == graph.v(e) && !allow_loops) continue;
   6.375 +    int rw = mwpfm.nodeValue(graph.u(e)) + mwpfm.nodeValue(graph.v(e))
   6.376 +      - weight[e] * mwpfm.dualScale;
   6.377 +
   6.378 +    check(rw >= 0, "Negative reduced weight");
   6.379 +    check(rw == 0 || !mwpfm.matching(e),
   6.380 +          "Non-zero reduced weight on matching edge");
   6.381 +  }
   6.382 +
   6.383 +  int pv = 0;
   6.384 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.385 +    int indeg = 0;
   6.386 +    for (InArcIt a(graph, n); a != INVALID; ++a) {
   6.387 +      if (mwpfm.matching(graph.source(a)) == a) {
   6.388 +        ++indeg;
   6.389 +      }
   6.390 +    }
   6.391 +    check(mwpfm.matching(n) != INVALID, "Invalid perfect matching");
   6.392 +    check(indeg == 1, "Invalid perfect matching");
   6.393 +    pv += weight[mwpfm.matching(n)];
   6.394 +    SmartGraph::Node o = graph.target(mwpfm.matching(n));
   6.395 +  }
   6.396 +
   6.397 +  int dv = 0;
   6.398 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.399 +    dv += mwpfm.nodeValue(n);
   6.400 +  }
   6.401 +
   6.402 +  check(pv * mwpfm.dualScale == dv * 2, "Wrong duality");
   6.403 +
   6.404 +  SmartGraph::NodeMap<bool> processed(graph, false);
   6.405 +  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {
   6.406 +    if (processed[n]) continue;
   6.407 +    processed[n] = true;
   6.408 +    if (mwpfm.matching(n) == INVALID) continue;
   6.409 +    int num = 1;
   6.410 +    Node v = graph.target(mwpfm.matching(n));
   6.411 +    while (v != n) {
   6.412 +      processed[v] = true;
   6.413 +      ++num;
   6.414 +      v = graph.target(mwpfm.matching(v));
   6.415 +    }
   6.416 +    check(num == 2 || num % 2 == 1, "Wrong cycle size");
   6.417 +    check(allow_loops || num != 1, "Wrong cycle size");
   6.418 +  }
   6.419 +
   6.420 +  return;
   6.421 +}
   6.422 +
   6.423 +
   6.424 +int main() {
   6.425 +
   6.426 +  for (int i = 0; i < lgfn; ++i) {
   6.427 +    SmartGraph graph;
   6.428 +    SmartGraph::EdgeMap<int> weight(graph);
   6.429 +
   6.430 +    istringstream lgfs(lgf[i]);
   6.431 +    graphReader(graph, lgfs).
   6.432 +      edgeMap("weight", weight).run();
   6.433 +
   6.434 +    bool perfect_with_loops;
   6.435 +    {
   6.436 +      MaxFractionalMatching<SmartGraph> mfm(graph, true);
   6.437 +      mfm.run();
   6.438 +      checkFractionalMatching(graph, mfm, true);
   6.439 +      perfect_with_loops = mfm.matchingSize() == countNodes(graph);
   6.440 +    }
   6.441 +
   6.442 +    bool perfect_without_loops;
   6.443 +    {
   6.444 +      MaxFractionalMatching<SmartGraph> mfm(graph, false);
   6.445 +      mfm.run();
   6.446 +      checkFractionalMatching(graph, mfm, false);
   6.447 +      perfect_without_loops = mfm.matchingSize() == countNodes(graph);
   6.448 +    }
   6.449 +
   6.450 +    {
   6.451 +      MaxFractionalMatching<SmartGraph> mfm(graph, true);
   6.452 +      bool result = mfm.runPerfect();
   6.453 +      checkPerfectFractionalMatching(graph, mfm, result, true);
   6.454 +      check(result == perfect_with_loops, "Wrong perfect matching");
   6.455 +    }
   6.456 +
   6.457 +    {
   6.458 +      MaxFractionalMatching<SmartGraph> mfm(graph, false);
   6.459 +      bool result = mfm.runPerfect();
   6.460 +      checkPerfectFractionalMatching(graph, mfm, result, false);
   6.461 +      check(result == perfect_without_loops, "Wrong perfect matching");
   6.462 +    }
   6.463 +
   6.464 +    {
   6.465 +      MaxWeightedFractionalMatching<SmartGraph> mwfm(graph, weight, true);
   6.466 +      mwfm.run();
   6.467 +      checkWeightedFractionalMatching(graph, weight, mwfm, true);
   6.468 +    }
   6.469 +
   6.470 +    {
   6.471 +      MaxWeightedFractionalMatching<SmartGraph> mwfm(graph, weight, false);
   6.472 +      mwfm.run();
   6.473 +      checkWeightedFractionalMatching(graph, weight, mwfm, false);
   6.474 +    }
   6.475 +
   6.476 +    {
   6.477 +      MaxWeightedPerfectFractionalMatching<SmartGraph> mwpfm(graph, weight,
   6.478 +                                                             true);
   6.479 +      bool perfect = mwpfm.run();
   6.480 +      check(perfect == (mwpfm.matchingSize() == countNodes(graph)),
   6.481 +            "Perfect matching found");
   6.482 +      check(perfect == perfect_with_loops, "Wrong perfect matching");
   6.483 +
   6.484 +      if (perfect) {
   6.485 +        checkWeightedPerfectFractionalMatching(graph, weight, mwpfm, true);
   6.486 +      }
   6.487 +    }
   6.488 +
   6.489 +    {
   6.490 +      MaxWeightedPerfectFractionalMatching<SmartGraph> mwpfm(graph, weight,
   6.491 +                                                             false);
   6.492 +      bool perfect = mwpfm.run();
   6.493 +      check(perfect == (mwpfm.matchingSize() == countNodes(graph)),
   6.494 +            "Perfect matching found");
   6.495 +      check(perfect == perfect_without_loops, "Wrong perfect matching");
   6.496 +
   6.497 +      if (perfect) {
   6.498 +        checkWeightedPerfectFractionalMatching(graph, weight, mwpfm, false);
   6.499 +      }
   6.500 +    }
   6.501 +
   6.502 +  }
   6.503 +
   6.504 +  return 0;
   6.505 +}