source: lemon/lemon/christofides_tsp.h @ 1270:dceba191c00d

/* -*- mode: C++; indent-tabs-mode: nil; -*-
 *
 * This file is a part of LEMON, a generic C++ optimization library.
 *
 * Copyright (C) 2003-2013
 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
 * (Egervary Research Group on Combinatorial Optimization, EGRES).
 *
 * Permission to use, modify and distribute this software is granted
 * provided that this copyright notice appears in all copies. For
 * precise terms see the accompanying LICENSE file.
 *
 * This software is provided "AS IS" with no warranty of any kind,
 * express or implied, and with no claim as to its suitability for any
 * purpose.
 *
 */

#ifndef LEMON_CHRISTOFIDES_TSP_H
#define LEMON_CHRISTOFIDES_TSP_H

/// \ingroup tsp
/// \file
/// \brief Christofides algorithm for symmetric TSP

#include <lemon/full_graph.h>
#include <lemon/smart_graph.h>
#include <lemon/kruskal.h>
#include <lemon/matching.h>
#include <lemon/euler.h>

namespace lemon {

  /// \ingroup tsp
  ///
  /// \brief Christofides algorithm for symmetric TSP.
  ///
  /// ChristofidesTsp implements Christofides' heuristic for solving
  /// symmetric \ref tsp "TSP".
  ///
  /// This a well-known approximation method for the TSP problem with
  /// metric cost function.
  /// It has a guaranteed approximation factor of 3/2 (i.e. it finds a tour
  /// whose total cost is at most 3/2 of the optimum), but it usually
  /// provides better solutions in practice.
  /// This implementation runs in O(n<sup>3</sup>log(n)) time.
  ///
  /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and
  /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching"
  /// in the subgraph induced by the nodes that have odd degree in the
  /// spanning tree.
  /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal"
  /// of the union of the spanning tree and the matching.
  /// During this last step, the algorithm simply skips the visited nodes
  /// (i.e. creates shortcuts) assuming that the triangle inequality holds
  /// for the cost function.
  ///
  /// \tparam CM Type of the cost map.
  ///
  /// \warning CM::Value must be a signed number type.
  template <typename CM>
  class ChristofidesTsp
  {
    public:

      /// Type of the cost map
      typedef CM CostMap;
      /// Type of the edge costs
      typedef typename CM::Value Cost;

    private:

      GRAPH_TYPEDEFS(FullGraph);

      const FullGraph &_gr;
      const CostMap &_cost;
      std::vector<Node> _path;
      Cost _sum;

    public:

      /// \brief Constructor
      ///
      /// Constructor.
      /// \param gr The \ref FullGraph "full graph" the algorithm runs on.
      /// \param cost The cost map.
      ChristofidesTsp(const FullGraph &gr, const CostMap &cost)
        : _gr(gr), _cost(cost) {}

      /// \name Execution Control
      /// @{

      /// \brief Runs the algorithm.
      ///
      /// This function runs the algorithm.
      ///
      /// \return The total cost of the found tour.
      Cost run() {
        _path.clear();

        if (_gr.nodeNum() == 0) return _sum = 0;
        else if (_gr.nodeNum() == 1) {
          _path.push_back(_gr(0));
          return _sum = 0;
        }
        else if (_gr.nodeNum() == 2) {
          _path.push_back(_gr(0));
          _path.push_back(_gr(1));
          return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))];
        }

        // Compute min. cost spanning tree
        std::vector<Edge> tree;
        kruskal(_gr, _cost, std::back_inserter(tree));

        FullGraph::NodeMap<int> deg(_gr, 0);
        for (int i = 0; i != int(tree.size()); ++i) {
          Edge e = tree[i];
          ++deg[_gr.u(e)];
          ++deg[_gr.v(e)];
        }

        // Copy the induced subgraph of odd nodes
        std::vector<Node> odd_nodes;
        for (NodeIt u(_gr); u != INVALID; ++u) {
          if (deg[u] % 2 == 1) odd_nodes.push_back(u);
        }

        SmartGraph sgr;
        SmartGraph::EdgeMap<Cost> scost(sgr);
        for (int i = 0; i != int(odd_nodes.size()); ++i) {
          sgr.addNode();
        }
        for (int i = 0; i != int(odd_nodes.size()); ++i) {
          for (int j = 0; j != int(odd_nodes.size()); ++j) {
            if (j == i) continue;
            SmartGraph::Edge e =
              sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j));
            scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])];
          }
        }

        // Compute min. cost perfect matching
        MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> >
          mwpm(sgr, scost);
        mwpm.run();

        for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) {
          if (mwpm.matching(e)) {
            tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))],
                                     odd_nodes[sgr.id(sgr.v(e))]) );
          }
        }

        // Join the spanning tree and the matching
        sgr.clear();
        for (int i = 0; i != _gr.nodeNum(); ++i) {
          sgr.addNode();
        }
        for (int i = 0; i != int(tree.size()); ++i) {
          int ui = _gr.id(_gr.u(tree[i])),
              vi = _gr.id(_gr.v(tree[i]));
          sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi));
        }

        // Compute the tour from the Euler traversal
        SmartGraph::NodeMap<bool> visited(sgr, false);
        for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) {
          SmartGraph::Node n = sgr.target(e);
          if (!visited[n]) {
            _path.push_back(_gr(sgr.id(n)));
            visited[n] = true;
          }
        }

        _sum = _cost[_gr.edge(_path.back(), _path.front())];
        for (int i = 0; i < int(_path.size())-1; ++i) {
          _sum += _cost[_gr.edge(_path[i], _path[i+1])];
        }

        return _sum;
      }

      /// @}

      /// \name Query Functions
      /// @{

      /// \brief The total cost of the found tour.
      ///
      /// This function returns the total cost of the found tour.
      ///
      /// \pre run() must be called before using this function.
      Cost tourCost() const {
        return _sum;
      }

      /// \brief Returns a const reference to the node sequence of the
      /// found tour.
      ///
      /// This function returns a const reference to a vector
      /// that stores the node sequence of the found tour.
      ///
      /// \pre run() must be called before using this function.
      const std::vector<Node>& tourNodes() const {
        return _path;
      }

      /// \brief Gives back the node sequence of the found tour.
      ///
      /// This function copies the node sequence of the found tour into
      /// an STL container through the given output iterator. The
      /// <tt>value_type</tt> of the container must be <tt>FullGraph::Node</tt>.
      /// For example,
      /// \code
      /// std::vector<FullGraph::Node> nodes(countNodes(graph));
      /// tsp.tourNodes(nodes.begin());
      /// \endcode
      /// or
      /// \code
      /// std::list<FullGraph::Node> nodes;
      /// tsp.tourNodes(std::back_inserter(nodes));
      /// \endcode
      ///
      /// \pre run() must be called before using this function.
      template <typename Iterator>
      void tourNodes(Iterator out) const {
        std::copy(_path.begin(), _path.end(), out);
      }

      /// \brief Gives back the found tour as a path.
      ///
      /// This function copies the found tour as a list of arcs/edges into
      /// the given \ref lemon::concepts::Path "path structure".
      ///
      /// \pre run() must be called before using this function.
      template <typename Path>
      void tour(Path &path) const {
        path.clear();
        for (int i = 0; i < int(_path.size()) - 1; ++i) {
          path.addBack(_gr.arc(_path[i], _path[i+1]));
        }
        if (int(_path.size()) >= 2) {
          path.addBack(_gr.arc(_path.back(), _path.front()));
        }
      }

      /// @}

  };

}; // namespace lemon

#endif