# 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.

Availability:

• Official on v3.0.0
• Renamed on v2.5.0
• Experimental on v2.3.0
• pgr_maxFlowBoykovKolmogorov

## 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 above.
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¶

edges_sql: an SQL query, 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_id BIGINT Identifier of the edge in the original query(edges_sql).
source BIGINT Identifier of the first end point vertex of the edge.
target BIGINT Identifier of the second end point vertex of the edge.
flow BIGINT Flow through the edge in the direction (source, target).
residual_capacity BIGINT Residual capacity of the edge in the direction (source, target).