push_relabel_max_flow

// named parameter version
template <class Graph, class P, class T, class R>
typename property_traits<CapacityEdgeMap>::value_type
push_relabel_max_flow(Graph& g,
   typename graph_traits<Graph>::vertex_descriptor src,
   typename graph_traits<Graph>::vertex_descriptor sink,
   const bgl_named_params<P, T, R>& params = all defaults)

// non-named parameter version
template <class Graph,
	  class CapacityEdgeMap, class ResidualCapacityEdgeMap,
	  class ReverseEdgeMap, class VertexIndexMap>
typename property_traits<CapacityEdgeMap>::value_type
push_relabel_max_flow(Graph& g,
   typename graph_traits<Graph>::vertex_descriptor src,
   typename graph_traits<Graph>::vertex_descriptor sink,
   CapacityEdgeMap cap, ResidualCapacityEdgeMap res,
   ReverseEdgeMap rev, VertexIndexMap index_map)

The push_relabel_max_flow() function calculates the maximum flow of a network. See Section Network Flow Algorithms for a description of maximum flow. The calculated maximum flow will be the return value of the function. The function also calculates the flow values f(u,v) for all (u,v) in E, which are returned in the form of the residual capacity r(u,v) = c(u,v) - f(u,v).

There are several special requirements on the input graph and property map parameters for this algorithm. First, the directed graph G=(V,E) that represents the network must be augmented to include the reverse edge for every edge in E. That is, the input graph should be G_in = (V,{E U E^T}). The ReverseEdgeMap argument rev must map each edge in the original graph to its reverse edge, that is (u,v) -> (v,u) for all (u,v) in E. The CapacityEdgeMap argument cap must map each edge in E to a positive number, and each edge in E^T to 0.

This algorithm was developed by Goldberg.

Complexity

Where Defined

boost/graph/push_relabel_max_flow.hpp

Parameters

A directed graph. The graph's type must be a model of Vertex List Graph. For each edge (u,v) in the graph, the reverse edge (v,u) must also be in the graph.

The source vertex for the flow network graph.

The sink vertex for the flow network graph.

Named Parameters

The edge capacity property map. The type must be a model of a constant Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
Default: get(edge_capacity, g)

The edge residual capacity property map. The type must be a model of a mutable Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
Default: get(edge_residual_capacity, g)

An edge property map that maps every edge (u,v) in the graph to the reverse edge (v,u). The map must be a model of constant Lvalue Property Map. The key type of the map must be the graph's edge descriptor type.
Default: get(edge_reverse, g)

Maps each vertex of the graph to a unique integer in the range [0, num_vertices(g)). The map must be a model of constant LvaluePropertyMap. The key type of the map must be the graph's vertex descriptor type.
Default: get(vertex_index, g) Note: if you use this default, make sure your graph has an internal vertex_index property. For example, adjacency_list with VertexList=listS does not have an internal vertex_index property.

Example

#include <boost/config.hpp>
#include <iostream>
#include <string>
#include <boost/graph/push_relabel_max_flow.hpp>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/read_dimacs.hpp>

int
main()
{
  using namespace boost;

  typedef adjacency_list_traits<vecS, vecS, directedS> Traits;
  typedef adjacency_list<vecS, vecS, directedS,
    property<vertex_name_t, std::string>,
    property<edge_capacity_t, long,
      property<edge_residual_capacity_t, long,
	property<edge_reverse_t, Traits::edge_descriptor> > >
  > Graph;

  Graph g;
  long flow;

  property_map<Graph, edge_capacity_t>::type
    capacity = get(edge_capacity, g);
  property_map<Graph, edge_reverse_t>::type
    rev = get(edge_reverse, g);
  property_map<Graph, edge_residual_capacity_t>::type
    residual_capacity = get(edge_residual_capacity, g);

  Traits::vertex_descriptor s, t;
  read_dimacs_max_flow(g, capacity, rev, s, t);

  flow = push_relabel_max_flow(g, s, t);

  std::cout << "c  The total flow:" << std::endl;
  std::cout << "s " << flow << std::endl << std::endl;

  std::cout << "c flow values:" << std::endl;
  graph_traits<Graph>::vertex_iterator u_iter, u_end;
  graph_traits<Graph>::out_edge_iterator ei, e_end;
  for (boost::tie(u_iter, u_end) = vertices(g); u_iter != u_end; ++u_iter)
    for (boost::tie(ei, e_end) = out_edges(*u_iter, g); ei != e_end; ++ei)
      if (capacity[*ei] > 0)
        std::cout << "f " << *u_iter << " " << target(*ei, g) << " "
                  << (capacity[*ei] - residual_capacity[*ei]) << std::endl;
  return 0;
}

c  The total flow:
s 4

c flow values:
f 0 1 4
f 1 2 4
f 2 3 2
f 2 4 2
f 3 1 0
f 3 6 2
f 4 5 3
f 5 6 0
f 5 7 3
f 6 4 1
f 6 7 1

See Also