RhodeCode

  ///

  /// The maximum weighted fractional matching is a relaxation of the

  /// maximum weighted matching problem where the odd set constraints

  /// are omitted.

  /// It can be formulated with the following linear program.

  /// \f[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\f]

  /// \f[x_e \ge 0\quad \forall e\in E\f]

  /// \f[\max \sum_{e\in E}x_ew_e\f]

  /// where \f$\delta(X)\f$ is the set of edges incident to a node in

  /// \f$X\f$. The result must be the union of a matching with one

  /// value edges and a set of odd length cycles with half value edges.

  ///

  /// The algorithm calculates an optimal fractional matching and a

  /// proof of the optimality. The solution of the dual problem can be

  /// used to check the result of the algorithm. The dual linear

  /// problem is the following.

  /// \f[ y_u + y_v \ge w_{uv} \quad \forall uv\in E\f]

  /// \f[y_u \ge 0 \quad \forall u \in V\f]

  /// \f[\min \sum_{u \in V}y_u \f]

  ///

  /// The algorithm can be executed with the run() function.

  /// After it the matching (the primal solution) and the dual solution

  /// can be obtained using the query functions.

  ///

  ///

  /// \tparam GR The undirected graph type the algorithm runs on.

  /// \tparam WM The type edge weight map. The default type is

  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename WM>

#else

  template <typename GR,

            typename WM = typename GR::template EdgeMap<int> >

#endif

  class MaxWeightedFractionalMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the edge weight map

    typedef WM WeightMap;

    /// The value type of the edge weights

    typedef typename WeightMap::Value Value;

    /// The type of the matching map

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

    /// \brief Scaling factor for primal solution

    ///

    /// \brief Scaling factor for dual solution

    ///

    /// Scaling factor for dual solution. It is equal to 4 or 1

    /// according to the value type.

    static const int dualScale =

      std::numeric_limits<Value>::is_integer ? 4 : 1;

  private:

    TEMPLATE_GRAPH_TYPEDEFS(Graph);

    typedef typename Graph::template NodeMap<Value> NodePotential;

    const Graph& _graph;

    const WeightMap& _weight;

    MatchingMap* _matching;

    NodePotential* _node_potential;

    int _node_num;

    bool _allow_loops;

    enum Status {

    }

    /// \brief Return the number of covered nodes in the matching.

    ///

    /// This function returns the number of covered nodes in the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    int matchingSize() const {

      int num = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_matching)[n] != INVALID) {

          ++num;

        }

      }

      return num;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// This function returns \c true if the given edge is in the

    /// found matching. The result is scaled by \ref primalScale

    /// "primal scale".

    ///

    /// \pre Either run() or start() must be called before using this function.

    }

    /// \brief Return the fractional matching arc (or edge) incident

    /// to the given node.

    ///

    /// This function returns one of the fractional matching arc (or

    /// edge) incident to the given node in the found matching or \c

    /// INVALID if the node is not covered by the matching or if the

    /// node is on an odd length cycle then it is the successor edge

    /// on the cycle.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Arc matching(const Node& node) const {

      return (*_matching)[node];

    }

    /// \brief Return a const reference to the matching map.

    ///

    /// This function returns a const reference to a node map that stores

    /// the matching arc (or edge) incident to each node.

    const MatchingMap& matchingMap() const {

      return *_matching;

    }

  /// matching algorithm. The implementation uses priority queues and

  /// provides \f$O(nm\log n)\f$ time complexity.

  ///

  /// The maximum weighted fractional perfect matching is a relaxation

  /// of the maximum weighted perfect matching problem where the odd

  /// set constraints are omitted.

  /// It can be formulated with the following linear program.

  /// \f[ \sum_{e \in \delta(u)}x_e = 1 \quad \forall u\in V\f]

  /// \f[x_e \ge 0\quad \forall e\in E\f]

  /// \f[\max \sum_{e\in E}x_ew_e\f]

  /// where \f$\delta(X)\f$ is the set of edges incident to a node in

  /// \f$X\f$. The result must be the union of a matching with one

  /// value edges and a set of odd length cycles with half value edges.

  ///

  /// The algorithm calculates an optimal fractional matching and a

  /// proof of the optimality. The solution of the dual problem can be

  /// used to check the result of the algorithm. The dual linear

  /// problem is the following.

  /// \f[ y_u + y_v \ge w_{uv} \quad \forall uv\in E\f]

  /// \f[\min \sum_{u \in V}y_u \f]

  ///

  /// The algorithm can be executed with the run() function.

  /// After it the matching (the primal solution) and the dual solution

  /// can be obtained using the query functions.

  ///

  /// \tparam GR The undirected graph type the algorithm runs on.

  /// \tparam WM The type edge weight map. The default type is

  /// \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename WM>

#else

  template <typename GR,

            typename WM = typename GR::template EdgeMap<int> >

#endif

  class MaxWeightedPerfectFractionalMatching {

  public:

    /// The graph type of the algorithm

    typedef GR Graph;

    /// The type of the edge weight map

    typedef WM WeightMap;

    /// The value type of the edge weights

    typedef typename WeightMap::Value Value;

    /// The type of the matching map

    typedef typename Graph::template NodeMap<typename Graph::Arc>

    MatchingMap;

    /// \brief Scaling factor for primal solution

    ///

    /// \brief Scaling factor for dual solution

    ///

    /// Scaling factor for dual solution. It is equal to 4 or 1

    /// according to the value type.

    static const int dualScale =

      std::numeric_limits<Value>::is_integer ? 4 : 1;

  private:

    TEMPLATE_GRAPH_TYPEDEFS(Graph);

    typedef typename Graph::template NodeMap<Value> NodePotential;

    const Graph& _graph;

    const WeightMap& _weight;

    MatchingMap* _matching;

    NodePotential* _node_potential;

    int _node_num;

    bool _allow_loops;

    enum Status {

    }

    /// \brief Return the number of covered nodes in the matching.

    ///

    /// This function returns the number of covered nodes in the matching.

    ///

    /// \pre Either run() or start() must be called before using this function.

    int matchingSize() const {

      int num = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        if ((*_matching)[n] != INVALID) {

          ++num;

        }

      }

      return num;

    }

    /// \brief Return \c true if the given edge is in the matching.

    ///

    /// This function returns \c true if the given edge is in the

    /// found matching. The result is scaled by \ref primalScale

    /// "primal scale".

    ///

    /// \pre Either run() or start() must be called before using this function.

    }

    /// \brief Return the fractional matching arc (or edge) incident

    /// to the given node.

    ///

    /// This function returns one of the fractional matching arc (or

    /// edge) incident to the given node in the found matching or \c

    /// INVALID if the node is not covered by the matching or if the

    /// node is on an odd length cycle then it is the successor edge

    /// on the cycle.

    ///

    /// \pre Either run() or start() must be called before using this function.

    Arc matching(const Node& node) const {

      return (*_matching)[node];

    }

    /// \brief Return a const reference to the matching map.

    ///

    /// This function returns a const reference to a node map that stores

    /// the matching arc (or edge) incident to each node.

    const MatchingMap& matchingMap() const {

      return *_matching;

    }

  const_mat_test.dualValue();

  const_mat_test.nodeValue(n);

}

void checkFractionalMatching(const SmartGraph& graph,

                             const MaxFractionalMatching<SmartGraph>& mfm,

                             bool allow_loops = true) {

  int pv = 0;

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    int indeg = 0;

    for (InArcIt a(graph, n); a != INVALID; ++a) {

      if (mfm.matching(graph.source(a)) == a) {

        ++indeg;

      }

    }

    if (mfm.matching(n) != INVALID) {

      check(indeg == 1, "Invalid matching");

      ++pv;

    } else {

      check(indeg == 0, "Invalid matching");

    }

  }

  check(pv == mfm.matchingSize(), "Wrong matching size");

  SmartGraph::NodeMap<bool> processed(graph, false);

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    if (processed[n]) continue;

    processed[n] = true;

    if (mfm.matching(n) == INVALID) continue;

    int num = 1;

    Node v = graph.target(mfm.matching(n));

    while (v != n) {

      processed[v] = true;

      ++num;

      v = graph.target(mfm.matching(v));

    }

    check(num == 2 || num % 2 == 1, "Wrong cycle size");

    check(allow_loops || num != 1, "Wrong cycle size");

  }

  int anum = 0, bnum = 0;

  SmartGraph::NodeMap<bool> neighbours(graph, false);

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    if (!mfm.barrier(n)) continue;

    ++anum;

    for (SmartGraph::InArcIt a(graph, n); a != INVALID; ++a) {

      Node u = graph.source(a);

      if (!allow_loops && u == n) continue;

      if (!neighbours[u]) {

        neighbours[u] = true;

        ++bnum;

      }

    }

  }

  check(anum - bnum + mfm.matchingSize() == countNodes(graph),

        "Wrong barrier");

}

void checkPerfectFractionalMatching(const SmartGraph& graph,

                             const MaxFractionalMatching<SmartGraph>& mfm,

                             bool perfect, bool allow_loops = true) {

  if (perfect) {

    for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

      int indeg = 0;

      for (InArcIt a(graph, n); a != INVALID; ++a) {

        if (mfm.matching(graph.source(a)) == a) {

          ++indeg;

        }

      }

      check(mfm.matching(n) != INVALID, "Invalid matching");

      check(indeg == 1, "Invalid matching");

    }

  } else {

    int anum = 0, bnum = 0;

    SmartGraph::NodeMap<bool> neighbours(graph, false);

    for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

      if (!mfm.barrier(n)) continue;

      ++anum;

      for (SmartGraph::InArcIt a(graph, n); a != INVALID; ++a) {

        Node u = graph.source(a);

        if (!allow_loops && u == n) continue;

        if (!neighbours[u]) {

          neighbours[u] = true;

          ++bnum;

        }

      }

    }

    check(anum - bnum > 0, "Wrong barrier");

  }

}

void checkWeightedFractionalMatching(const SmartGraph& graph,

                   const SmartGraph::EdgeMap<int>& weight,

                   const MaxWeightedFractionalMatching<SmartGraph>& mwfm,

                   bool allow_loops = true) {

  for (SmartGraph::EdgeIt e(graph); e != INVALID; ++e) {

    check(rw == 0 || !mwfm.matching(e),

          "Non-zero reduced weight on matching edge");

  }

  int pv = 0;

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    int indeg = 0;

    for (InArcIt a(graph, n); a != INVALID; ++a) {

      if (mwfm.matching(graph.source(a)) == a) {

        ++indeg;

      }

    }

    check(indeg <= 1, "Invalid matching");

    if (mwfm.matching(n) != INVALID) {

      check(mwfm.nodeValue(n) >= 0, "Invalid node value");

      check(indeg == 1, "Invalid matching");

      pv += weight[mwfm.matching(n)];

      SmartGraph::Node o = graph.target(mwfm.matching(n));

    } else {

      check(mwfm.nodeValue(n) == 0, "Invalid matching");

      check(indeg == 0, "Invalid matching");

    }

  }

  int dv = 0;

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    dv += mwfm.nodeValue(n);

  }

  check(pv * mwfm.dualScale == dv * 2, "Wrong duality");

  SmartGraph::NodeMap<bool> processed(graph, false);

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    if (processed[n]) continue;

    processed[n] = true;

    if (mwfm.matching(n) == INVALID) continue;

    int num = 1;

    Node v = graph.target(mwfm.matching(n));

    while (v != n) {

      processed[v] = true;

      ++num;

      v = graph.target(mwfm.matching(v));

    }

    check(num == 2 || num % 2 == 1, "Wrong cycle size");

    check(allow_loops || num != 1, "Wrong cycle size");

  }

  return;

  for (SmartGraph::EdgeIt e(graph); e != INVALID; ++e) {

    if (graph.u(e) == graph.v(e) && !allow_loops) continue;

    int rw = mwpfm.nodeValue(graph.u(e)) + mwpfm.nodeValue(graph.v(e))

      - weight[e] * mwpfm.dualScale;

    check(rw >= 0, "Negative reduced weight");

    check(rw == 0 || !mwpfm.matching(e),

          "Non-zero reduced weight on matching edge");

  }

  int pv = 0;

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    int indeg = 0;

    for (InArcIt a(graph, n); a != INVALID; ++a) {

      if (mwpfm.matching(graph.source(a)) == a) {

        ++indeg;

      }

    }

    check(mwpfm.matching(n) != INVALID, "Invalid perfect matching");

    check(indeg == 1, "Invalid perfect matching");

    pv += weight[mwpfm.matching(n)];

    SmartGraph::Node o = graph.target(mwpfm.matching(n));

  }

  int dv = 0;

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    dv += mwpfm.nodeValue(n);

  }

  check(pv * mwpfm.dualScale == dv * 2, "Wrong duality");

  SmartGraph::NodeMap<bool> processed(graph, false);

  for (SmartGraph::NodeIt n(graph); n != INVALID; ++n) {

    if (processed[n]) continue;

    processed[n] = true;

    if (mwpfm.matching(n) == INVALID) continue;

    int num = 1;

    Node v = graph.target(mwpfm.matching(n));

    while (v != n) {

      processed[v] = true;

      ++num;

      v = graph.target(mwpfm.matching(v));

    }

    check(num == 2 || num % 2 == 1, "Wrong cycle size");

    check(allow_loops || num != 1, "Wrong cycle size");

  }

  return;