# pgRouting Manual (3.6)

## pgr_maxFlow

# `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.

Boost Graph Inside

Availability

- Version 3.2.0

  - New proposed signature

    - `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, start vid, end vid)
pgr_maxFlow(Edges SQL, start vid, end vids)
pgr_maxFlow(Edges SQL, start vids, end vid)
pgr_maxFlow(Edges SQL, start vids, end vids)
pgr_maxFlow(Edges SQL, Combinations SQL)
RETURNS `BIGINT`

### One to One

pgr_maxFlow(Edges SQL, start vid, end vid)
RETURNS `BIGINT`

Example:

  From vertex \(11\) to vertex \(12\)

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      11, 12);
     pgr_maxflow
    -------------
             230
    (1 row)

### One to Many

pgr_maxFlow(Edges SQL, start vid, end vids)
RETURNS `BIGINT`

Example:

  From vertex \(11\) to vertices \(\{5, 10, 12\}\)

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      11, ARRAY[5, 10, 12]);
     pgr_maxflow
    -------------
             340
    (1 row)

### Many to One

pgr_maxFlow(Edges SQL, start vids, end vid)
RETURNS `BIGINT`

Example:

  From vertices \(\{11, 3, 17\}\) to vertex \(12\)

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      ARRAY[11, 3, 17], 12);
     pgr_maxflow
    -------------
             230
    (1 row)

### Many to Many

pgr_maxFlow(Edges SQL, start vids, end vids)
RETURNS `BIGINT`

Example:

  From vertices \(\{11, 3, 17\}\) to vertices \(\{5, 10, 12\}\)

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      ARRAY[11, 3, 17], ARRAY[5, 10, 12]);
     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 \(\{5, 6\}\) to vertices \(\{10, 15, 14\}\).

The combinations table:

    SELECT source, target FROM combinations
    WHERE target NOT IN (5, 6);
     source | target
    --------+--------
          5 |     10
          6 |     15
          6 |     14
    (3 rows)

The query:

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      'SELECT * FROM combinations WHERE target NOT IN (5, 6)');
     pgr_maxflow
    -------------
              80
    (1 row)

## Parameters

Column | Type | Description
--- | --- | ---
Edges SQL | `TEXT` | Edges SQL as described below
Combinations SQL | `TEXT` | Combinations SQL as described below
start vid | `BIGINT` | Identifier of the starting vertex of the path.
start vids | `ARRAY[BIGINT]` | Array of identifiers of starting vertices.
end vid | `BIGINT` | Identifier of the ending vertex of the path.
end vids | `ARRAY[BIGINT]` | Array of identifiers of ending vertices.

## Inner Queries

### Edges SQL

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`)

`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`

ANY-NUMERICAL:

  `SMALLINT`, `INTEGER`, `BIGINT`, `REAL`, `FLOAT`

### Combinations SQL

Parameter | Type | Description
--- | --- | ---
`source` | ANY-INTEGER | Identifier of the departure vertex.
`target` | ANY-INTEGER | Identifier of the arrival vertex.

Where:

ANY-INTEGER:

  `SMALLINT`, `INTEGER`, `BIGINT`

## Result columns

Type | Description
--- | ---
`BIGINT` | Maximum flow possible from the source(s) to the target(s)

## Additional Examples

Example:

  Manually assigned vertex combinations.

    SELECT * FROM pgr_maxFlow(
      'SELECT id, source, target, capacity, reverse_capacity
      FROM edges',
      'SELECT * FROM (VALUES (5, 10), (6, 15), (6, 14)) AS t(source, target)');
     pgr_maxflow
    -------------
              80
    (1 row)

## See Also

- Flow - Family of functions

  - pgr_pushRelabel

- https://www.boost.org/libs/graph/doc/push_relabel_max_flow.html

- https://en.wikipedia.org/wiki/Push%E2%80%93relabel_maximum_flow_algorithm

Indices and tables

- Index

- Search Page