# `edmonds_karp_max_flow`

    // named paramter version
    template <class Graph, class P, class T, class R>
    typename detail::edge_capacity_value<Graph, P, T, R>::value_type
    edmonds_karp_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 ColorMap, class PredEdgeMap>
    typename property_traits<CapacityEdgeMap>::value_type
    edmonds_karp_max_flow(Graph& g,
       typename graph_traits<Graph>::vertex_descriptor src,
       typename graph_traits<Graph>::vertex_descriptor sink,
       CapacityEdgeMap cap, ResidualCapacityEdgeMap res, ReverseEdgeMap rev,
       ColorMap color, PredEdgeMap pred)

The `edmonds_karp_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 Gin = (V,{E U ET}). 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 ET to 0.

The algorithm is due to Edmonds and Karp, though we are using the variation called the ``labeling algorithm'' described in Network Flows.

This algorithm provides a very simple and easy to implement solution to the maximum flow problem. However, there are several reasons why this algorithm is not as good as the `push_relabel_max_flow()` or the `kolmogorov_max_flow()` algorithm.

- In the non-integer capacity case, the time complexity is O(V E2) which is worse than the time complexity of the push-relabel algorithm O(V2E1/2) for all but the sparsest of graphs.
- In the integer capacity case, if the capacity bound U is very large then the algorithm will take a long time.

### Where Defined

`boost/graph/edmonds_karp_max_flow.hpp`

### Parameters

IN: `Graph& g`

> A directed graph. The graph's type must be a model of VertexListGraph and IncidenceGraph For each edge (u,v) in the graph, the reverse edge (v,u) must also be in the graph.

IN: `vertex_descriptor src`

> The source vertex for the flow network graph.

IN: `vertex_descriptor sink`

> The sink vertex for the flow network graph.

### Named Parameters

IN: `capacity_map(CapacityEdgeMap cap)`

> 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)`

OUT: `residual_capacity_map(ResidualCapacityEdgeMap res)`

> This maps edges to their residual capacity. 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)`

IN: `reverse_edge_map(ReverseEdgeMap rev)`

> 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)`

UTIL: `color_map(ColorMap color)`

> Used by the algorithm to keep track of progress during the breadth-first search stage. At the end of the algorithm, the white vertices define the minimum cut set. The map must be a model of mutable Lvalue Property Map. The key type of the map should be the graph's vertex descriptor type, and the value type must be a model of ColorValue.
> Default: an `iterator_property_map` created from a `std::vector` of `default_color_type` of size `num_vertices(g)` and using the `i_map` for the index map.

UTIL: `predecessor_map(PredEdgeMap pred)`

> Use by the algorithm to store augmenting paths. The map must be a model of mutable Lvalue Property Map. The key type must be the graph's vertex descriptor type and the value type must be the graph's edge descriptor type.
> Default: an `iterator_property_map` created from a `std::vector` of edge descriptors of size `num_vertices(g)` and using the `i_map` for the index map.

IN: `vertex_index_map(VertexIndexMap i_map)`

> Maps each vertex of the graph to a unique integer in the range `[0, num_vertices(g))`. This property map is only needed if the default for the color or predecessor map is used. The vertex index map must be a model of Readable Property Map. 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, `adjacenty_list` with `VertexList=listS` does not have an internal `vertex_index` property.

### Complexity

The time complexity is O(V E2) in the general case or O(V E U) if capacity values are integers bounded by some constant U.

### Example

The program in `example/edmonds-karp-eg.cpp` reads an example maximum flow problem (a graph with edge capacities) from a file in the DIMACS format and computes the maximum flow.

### See Also

`push_relabel_max_flow()`
`kolmogorov_max_flow()`.

Copyright © 2000-2001 | Jeremy Siek, Indiana University (jsiek@osl.iu.edu)