# maxflow

Maximum flow in graph

## Syntax

## Description

returns the maximum flow between
nodes `mf`

= maxflow(`G`

,`s,t`

)`s`

and `t`

. If graph `G`

is unweighted (that is, `G.Edges`

does not contain the variable
`Weight`

), then `maxflow`

treats all graph
edges as having a weight equal to 1.

## Examples

### Maximum Flow in Graph

Create and plot a weighted graph. The weighted edges represent flow capacities.

s = [1 1 2 2 3 4 4 4 5 5]; t = [2 3 3 4 5 3 5 6 4 6]; weights = [0.77 0.44 0.67 0.75 0.89 0.90 2 0.76 1 1]; G = digraph(s,t,weights); plot(G,'EdgeLabel',G.Edges.Weight,'Layout','layered');

Determine the maximum flow from node 1 to node 6.

mf = maxflow(G,1,6)

mf = 1.2100

### Maximum Flow with Specified Algorithm

Create and plot a graph. The weighted edges represent flow capacities.

```
s = [1 1 2 2 3 3 4];
t = [2 3 3 4 4 5 5];
weights = [10 6 15 5 10 3 8];
G = digraph(s,t,weights);
H = plot(G,'EdgeLabel',G.Edges.Weight);
```

Find the maximum flow value between node 1 and node 5. Specify `'augmentpath'`

to use the Ford-Fulkerson algorithm, and use two outputs to return a graph of the nonzero flows.

`[mf,GF] = maxflow(G,1,5,'augmentpath')`

mf = 11

GF = digraph with properties: Edges: [6x2 table] Nodes: [5x0 table]

Highlight and label the graph of nonzero flows.

H.EdgeLabel = {}; highlight(H,GF,'EdgeColor','r','LineWidth',2); st = GF.Edges.EndNodes; labeledge(H,st(:,1),st(:,2),GF.Edges.Weight);

### Minimum Cut Computation

Create and plot a weighted graph. The edge weights represent flow capacities.

s = [1 1 2 3 3 4 4 5 5]; t = [2 3 3 2 5 5 6 4 6]; weights = [0.77 0.44 0.67 0.69 0.73 2 0.78 1 1]; G = digraph(s,t,weights); plot(G,'EdgeLabel',G.Edges.Weight,'Layout','layered')

Find the maximum flow and minimum cut of the graph.

[mf,~,cs,ct] = maxflow(G,1,6)

mf = 0.7300

`cs = `*3×1*
1
2
3

`ct = `*3×1*
4
5
6

Plot the minimum cut, using the `cs`

nodes as sources and the `ct`

nodes as sinks. Highlight the `cs`

nodes as red and the `ct`

nodes as green. Note that the weight of the edge that connects these two sets of nodes is equal to the maximum flow.

H = plot(G,'Layout','layered','Sources',cs,'Sinks',ct, ... 'EdgeLabel',G.Edges.Weight); highlight(H,cs,'NodeColor','red') highlight(H,ct,'NodeColor','green')

## Input Arguments

`s,t`

— Node pair (as separate arguments)

node indices | node names

Node pair, specified as separate arguments of node indices or node names to indicate the source node and target node. This table shows the different ways to refer to nodes either by their node indices or by their node names.

Value | Example |
---|---|

Scalar node index | `1` |

Character vector node name | `'A'` |

String scalar node name | `"A"` |

**Example: **`mf = maxflow(G,'A','B')`

**Example: **`mf = maxflow(G,1,10)`

**Data Types: **`double`

| `char`

| `string`

`algorithm`

— Maximum flow algorithm

`'searchtrees'`

(default) | `'augmentpath'`

| `'pushrelabel'`

Maximum flow algorithm, specified as one of the entries in the table.

**Note**

You can only specify nondefault `algorithm`

options
with a directed graph.

Option | Description |
---|---|

`'searchtrees'` (default) |
Uses the Boykov-Kolmogorov algorithm. Computes the
maximum flow by constructing two search trees associated
with nodes |

`'augmentpath'` |
Uses the Ford-Fulkerson algorithm. Computes the maximum flow iteratively by finding an augmenting path in a residual directed graph. The directed graph cannot have any parallel edges of
opposite direction between the same two nodes, unless
the weight of one of those edges is zero. So if the
graph contains edge |

`'pushrelabel'` |
Computes the maximum flow by pushing a node's excess flow to its neighbors and then relabeling the node. The directed graph cannot have any parallel edges of
opposite direction between the same two nodes, unless
the weight of one of those edges is zero. So if the
graph contains edge |

**Example: **```
mf =
maxflow(G,'A','D','augmentpath')
```

## Output Arguments

`mf`

— Maximum flow

scalar

Maximum flow, returned as a scalar.

`GF`

— Directed graph of flows

`digraph`

object

Directed graph of flows, returned as a `digraph`

object.
`GF`

contains the same nodes as `G`

,
but only contains those edges of `G`

that have a nonzero
flow. For multigraphs with multiple edges between the same two nodes,
`GF`

contains a single edge reflecting the flow through
the multiple edges.

`cs`

— Minimum cut source node IDs

node indices | node names

Minimum cut source node IDs, returned as node indices or node names.

If

`s`

and`t`

specify numeric node indices, then`cs`

and`ct`

also contain node indices.If

`s`

and`t`

specify node names, then`cs`

and`ct`

also contain node names.

`ct`

— Minimum cut target node IDs

scalar | vector | character vector | cell array of character vectors

Minimum cut target node IDs, returned as node indices or node names.

If

`s`

and`t`

specify numeric node indices, then`cs`

and`ct`

also contain node indices.If

`s`

and`t`

specify node names, then`cs`

and`ct`

also contain node names.

## More About

### Maximum Flow

In the context of maximum flow, the edges in a graph are
considered to have a *capacity* as represented by the edge
weight. The capacity of an edge is the amount of flow that can pass through that
edge. Therefore, the maximum flow between two nodes in a graph maximizes the amount
of flow passing from the source node, `s`

, to the target node,
`t`

, based on the capacities of the connecting edges.

### Minimum Cut

A minimum cut partitions the directed graph nodes into two sets,
`cs`

and `ct`

, such that the sum of the
weights of all edges connecting `cs`

and `ct`

(weight of the cut) is minimized. The weight of the minimum cut is equal to the
maximum flow value, `mf`

.

The entries in `cs`

and `ct`

indicate the nodes
of `G`

associated with nodes `s`

and
`t`

, respectively. `cs`

and
`ct`

satisfy ```
numel(cs) + numel(ct) =
numnodes(G)
```

.

## Version History

**Introduced in R2015b**

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