# 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.

Boost Graph Inside

Availability:

• Version 3.2.0

• New proposed function:

• pgr_boykovKolmogorov(Combinations)

• Version 3.0.0

• Official function

• Version 2.5.0

• Renamed from pgr_maxFlowBoykovKolmogorov

• Proposed function

• Version 2.3.0

• New Experimental function

## 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)
pgr_boykovKolmogorov(Edges SQL, Combinations SQL) -- Proposed on v3.2
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)



### Combinations¶

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

Example

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

SELECT * FROM pgr_boykovKolmogorov(
'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)'
);
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

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.

## 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).