# pgr_maxFlow¶

pgr_maxFlow — Calculates the maximum flow in a directed graph from the source(s) to the targets(s) using the Push Relabel algorithm.

Availability

• Version 3.2.0

• New proposed function:

• pgr_maxFlow(Combinations)

• Version 3.0.0

• Official function

• Version 2.4.0

• New Proposed function

## Description¶

The main characteristics are:

• The graph is directed.

• Calculates the maximum flow from the source(s) to the target(s).

• When the maximum flow is 0 then there is no flow and 0 is returned.

• There is no flow when a source is the same as a target.

• Any duplicated value in the source(s) or target(s) are ignored.

• Uses the pgr_pushRelabel algorithm.

• Running time: $$O( V ^ 3)$$

## Signatures¶

Summary

pgr_maxFlow(Edges SQL, source,  target)
pgr_maxFlow(Edges SQL, sources,  target)
pgr_maxFlow(Edges SQL, source,  targets)
pgr_maxFlow(Edges SQL, sources,  targets)
pgr_maxFlow(Edges SQL, Combinations SQL)
RETURNS BIGINT


### One to One¶

pgr_maxFlow(Edges SQL, source,  target)
RETURNS BIGINT

Example

From vertex $$6$$ to vertex $$11$$

SELECT * FROM pgr_maxFlow(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table'
, 6, 11
);
pgr_maxflow
-------------
230
(1 row)



### One to Many¶

pgr_maxFlow(Edges SQL, source,  targets)
RETURNS BIGINT

Example

From vertex $$6$$ to vertices $$\{11, 1, 13\}$$

SELECT * FROM pgr_maxFlow(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table'
, 6, ARRAY[11, 1, 13]
);
pgr_maxflow
-------------
340
(1 row)



### Many to One¶

pgr_maxFlow(Edges SQL, sources,  target)
RETURNS BIGINT

Example

From vertices $$\{6, 8, 12\}$$ to vertex $$11$$

SELECT * FROM pgr_maxFlow(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table'
, ARRAY[6, 8, 12], 11
);
pgr_maxflow
-------------
230
(1 row)



### Many to Many¶

pgr_maxFlow(Edges SQL, sources,  targets)
RETURNS BIGINT

Example

From vertices $$\{6, 8, 12\}$$ to vertices $$\{1, 3, 11\}$$

SELECT * FROM pgr_maxFlow(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table'
, ARRAY[6, 8, 12], ARRAY[1, 3, 11]
);
pgr_maxflow
-------------
360
(1 row)



### Combinations¶

pgr_maxFlow(Edges SQL, Combinations SQL)
RETURNS BIGINT

Example

Using a combinations table, equivalent to calculating result from vertices $$\{6, 8, 12\}$$ to vertices $$\{1, 3, 11\}$$.

SELECT * FROM pgr_maxFlow(
'SELECT id,
source,
target,
capacity,
reverse_capacity
FROM edge_table',
'SELECT * FROM ( VALUES (6, 1), (8, 3), (12, 11), (8, 1) ) AS t(source, target)'
);
pgr_maxflow
-------------
360
(1 row)



## Parameters¶

Column

Type

Default

Description

Edges SQL

TEXT

Edges query as described in Inner Queries.

Combinations SQL

TEXT

Combinations query as described in Inner Queries.

source

BIGINT

Identifier of the starting vertex of the flow.

sources

ARRAY[BIGINT]

Array of identifiers of the starting vertices of the flow.

target

BIGINT

Identifier of the ending vertex of the flow.

targets

ARRAY[BIGINT]

Array of identifiers of the ending vertices of the flow.

## Inner queries¶

Edges SQL

an SQL query of a directed graph of capacities, which should return a set of rows with the following columns:

Column

Type

Default

Description

id

ANY-INTEGER

Identifier of the edge.

source

ANY-INTEGER

Identifier of the first end point vertex of the edge.

target

ANY-INTEGER

Identifier of the second end point vertex of the edge.

capacity

ANY-INTEGER

Weight of the edge (source, target)

• When negative: edge (source, target) does not exist, therefore it’s not part of the graph.

reverse_capacity

ANY-INTEGER

-1

Weight of the edge (target, source),

• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

Combinations SQL

an SQL query which should return a set of rows with the following columns:

Column

Type

Default

Description

source

ANY-INTEGER

Identifier of the first end point vertex of the edge.

target

ANY-INTEGER

Identifier of the second end point vertex of the edge.

Where:

ANY-INTEGER

SMALLINT, INTEGER, BIGINT

The function aggregates the sources and the targets, removes the duplicates, and then it calculates the result from the resultant source vertices to the target vertices.

## Return Columns¶

Type

Description

BIGINT

Maximum flow possible from the source(s) to the target(s)