pgr_boykovKolmogorov

pgr_boykovKolmogorov — Calculates the flow on the graph edges that maximizes the flow from the sources to the targets using Boykov Kolmogorov algorithm.

_images/boost-inside.jpeg

Boost Graph Inside

Availability:

  • Version 3.0.0

    • Official function

  • Version 2.5.0

    • Renamed from pgr_maxFlowBoykovKolmogorov

    • Proposed function

  • Version 2.3.0

    • New Experimental function

Support

Description

The main characteristics are:

  • The graph is directed.

  • Process is done only on edges with positive capacities.

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

  • Calculates the flow/residual capacity for each edge. In the output

    • Edges with zero flow are omitted.

  • Creates a super source and edges to all the source(s), and a super target and the edges from all the targets(s).

  • The maximum flow through the graph is guaranteed to be the value returned by pgr_maxFlow when executed with the same parameters and can be calculated:

    • By aggregation of the outgoing flow from the sources

    • By aggregation of the incoming flow to the targets

  • Running time: Polynomial

Signatures

Summary

pgr_boykovKolmogorov(Edges SQL, source,  target)
pgr_boykovKolmogorov(Edges SQL, sources, target)
pgr_boykovKolmogorov(Edges SQL, source,  targets)
pgr_boykovKolmogorov(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET

One to One

pgr_boykovKolmogorov(Edges SQL, source,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example

From vertex \(6\) to vertex \(11\)

SELECT * FROM pgr_boykovKolmogorov(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , 6, 11
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |   10 |         5 |      10 |  100 |                30
   2 |    8 |         6 |       5 |  100 |                30
   3 |   11 |         6 |      11 |  130 |                 0
   4 |   12 |        10 |      11 |  100 |                 0
(4 rows)

One to Many

pgr_boykovKolmogorov(Edges SQL, source,  targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example

From vertex \(6\) to vertices \(\{1, 3, 11\}\)

SELECT * FROM pgr_boykovKolmogorov(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , 6, ARRAY[1, 3, 11]
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |    1 |         2 |       1 |   50 |                80
   2 |    3 |         4 |       3 |   80 |                50
   3 |    4 |         5 |       2 |   50 |                 0
   4 |   10 |         5 |      10 |   80 |                50
   5 |    8 |         6 |       5 |  130 |                 0
   6 |    9 |         6 |       9 |   80 |                50
   7 |   11 |         6 |      11 |  130 |                 0
   8 |   16 |         9 |       4 |   80 |                 0
   9 |   12 |        10 |      11 |   80 |                20
(9 rows)

Many to One

pgr_boykovKolmogorov(Edges SQL, sources,  target)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example

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

SELECT * FROM pgr_boykovKolmogorov(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , ARRAY[6, 8, 12], 11
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |   10 |         5 |      10 |  100 |                30
   2 |    8 |         6 |       5 |  100 |                30
   3 |   11 |         6 |      11 |  130 |                 0
   4 |   12 |        10 |      11 |  100 |                 0
(4 rows)

Many to Many

pgr_boykovKolmogorov(Edges SQL, sources,  targets)
RETURNS SET OF (seq, edge, start_vid, end_vid, flow, residual_capacity)
OR EMPTY SET
Example

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

SELECT * FROM pgr_boykovKolmogorov(
    'SELECT id,
            source,
            target,
            capacity,
            reverse_capacity
    FROM edge_table'
    , ARRAY[6, 8, 12], ARRAY[1, 3, 11]
);
 seq | edge | start_vid | end_vid | flow | residual_capacity
-----+------+-----------+---------+------+-------------------
   1 |    1 |         2 |       1 |   50 |                80
   2 |    3 |         4 |       3 |   80 |                50
   3 |    4 |         5 |       2 |   50 |                 0
   4 |   10 |         5 |      10 |  100 |                30
   5 |    8 |         6 |       5 |  130 |                 0
   6 |    9 |         6 |       9 |   80 |                50
   7 |   11 |         6 |      11 |  130 |                 0
   8 |    7 |         8 |       5 |   20 |                30
   9 |   16 |         9 |       4 |   80 |                 0
  10 |   12 |        10 |      11 |  100 |                 0
(10 rows)

Parameters

Column

Type

Default

Description

Edges SQL

TEXT

The edges SQL query as described in Inner Query.

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 query

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

Result Columns

Column

Type

Description

seq

INT

Sequential value starting from 1.

edge

BIGINT

Identifier of the edge in the original query(edges_sql).

start_vid

BIGINT

Identifier of the first end point vertex of the edge.

end_vid

BIGINT

Identifier of the second end point vertex of the edge.

flow

BIGINT

Flow through the edge in the direction (start_vid, end_vid).

residual_capacity

BIGINT

Residual capacity of the edge in the direction (start_vid, end_vid).