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

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