# Documentation

### This is machine translation

Translated by
Mouseover text to see original. Click the button below to return to the English version of the page.

# graphmaxflow

Calculate maximum flow in directed graph

## Syntax

[MaxFlow, FlowMatrix, Cut] = graphmaxflow(G, SNode, TNode)
[...] = graphmaxflow(G, SNode, TNode, ...'Capacity', CapacityValue, ...)
[...] = graphmaxflow(G, SNode, TNode, ...'Method', MethodValue, ...)

## Arguments

 G N-by-N sparse matrix that represents a directed graph. Nonzero entries in matrix G represent the capacities of the edges. SNode Node in G. TNode Node in G. CapacityValue Column vector that specifies custom capacities for the edges in matrix G. It must have one entry for every nonzero value (edge) in matrix G. The order of the custom capacities in the vector must match the order of the nonzero values in matrix G when it is traversed column-wise. By default, graphmaxflow gets capacity information from the nonzero entries in matrix G. MethodValue Character vector that specifies the algorithm used to find the minimal spanning tree (MST). Choices are:'Edmonds' — Uses the Edmonds and Karp algorithm, the implementation of which is based on a variation called the labeling algorithm. Time complexity is O(N*E^2), where N and E are the number of nodes and edges respectively.'Goldberg' — Default algorithm. Uses the Goldberg algorithm, which uses the generic method known as preflow-push. Time complexity is O(N^2*sqrt(E)), where N and E are the number of nodes and edges respectively.

## Description

### Tip

For introductory information on graph theory functions, see Graph Theory Functions.

[MaxFlow, FlowMatrix, Cut] = graphmaxflow(G, SNode, TNode)calculates the maximum flow of directed graph G from node SNode to node TNode. Input G is an N-by-N sparse matrix that represents a directed graph. Nonzero entries in matrix G represent the capacities of the edges. Output MaxFlow is the maximum flow, and FlowMatrix is a sparse matrix with all the flow values for every edge. FlowMatrix(X,Y) is the flow from node X to node Y. Output Cut is a logical row vector indicating the nodes connected to SNode after calculating the minimum cut between SNode and TNode. If several solutions to the minimum cut problem exist, then Cut is a matrix.

### Tip

The algorithm that determines Cut, all minimum cuts, has a time complexity of O(2^N), where N is the number of nodes. If this information is not needed, use the graphmaxflow function without the third output.

[...] = graphmaxflow(G, SNode, TNode, ...'PropertyName', PropertyValue, ...) calls graphmaxflow with optional properties that use property name/property value pairs. You can specify one or more properties in any order. Each PropertyName must be enclosed in single quotes and is case insensitive. These property name/property value pairs are as follows:

[...] = graphmaxflow(G, SNode, TNode, ...'Capacity', CapacityValue, ...) lets you specify custom capacities for the edges. CapacityValue is a column vector having one entry for every nonzero value (edge) in matrix G. The order of the custom capacities in the vector must match the order of the nonzero values in matrix G when it is traversed column-wise. By default, graphmaxflow gets capacity information from the nonzero entries in matrix G.

[...] = graphmaxflow(G, SNode, TNode, ...'Method', MethodValue, ...) lets you specify the algorithm used to find the minimal spanning tree (MST). Choices are:

• 'Edmonds' — Uses the Edmonds and Karp algorithm, the implementation of which is based on a variation called the labeling algorithm. Time complexity is O(N*E^2), where N and E are the number of nodes and edges respectively.

• 'Goldberg' — Default algorithm. Uses the Goldberg algorithm, which uses the generic method known as preflow-push. Time complexity is O(N^2*sqrt(E)), where N and E are the number of nodes and edges respectively.

## Examples

collapse all

This example shows how to calculate the maximum flow in a directed graph.

Create a directed graph with six nodes and eight edges.

cm = sparse([1 1 2 2 3 3 4 5],[2 3 4 5 4 5 6 6],...
[2 3 3 1 1 1 2 3],6,6)
cm =

(1,2)        2
(1,3)        3
(2,4)        3
(3,4)        1
(2,5)        1
(3,5)        1
(4,6)        2
(5,6)        3

Calculate the maximum flow in the graph from node 1 to node 6.

[M,F,K] = graphmaxflow(cm,1,6)
M =

4

F =

(1,2)        2
(1,3)        2
(2,4)        1
(3,4)        1
(2,5)        1
(3,5)        1
(4,6)        2
(5,6)        2

K =

2x6 logical array

1   1   1   1   0   0
1   0   1   0   0   0

Notice that K is a two-row matrix because there are two possible solutions to the minimum cut problem.

View the graph with the original capacities.

h = view(biograph(cm,[],'ShowWeights','on'))
Biograph object with 6 nodes and 8 edges.

View the graph with the calculated maximum flows.

view(biograph(F,[],'ShowWeights','on'))

Show one solution to the minimum cut problem in the original graph.

set(h.Nodes(K(1,:)),'Color',[1 0 0])

## References

[1] Edmonds, J. and Karp, R.M. (1972). Theoretical improvements in the algorithmic efficiency for network flow problems. Journal of the ACM 19, 248-264.

[2] Goldberg, A.V. (1985). A New Max-Flow Algorithm. MIT Technical Report MIT/LCS/TM-291, Laboratory for Computer Science, MIT.

[3] Siek, J.G., Lee, L-Q, and Lumsdaine, A. (2002). The Boost Graph Library User Guide and Reference Manual, (Upper Saddle River, NJ:Pearson Education).