pgr_maxFlowMinCost - Experimental¶

pgr_maxFlowMinCost — Calculates the flow on the graph edges that maximizes the flow and minimizes the cost from the sources to the targets.

Boost Graph Inside

Warning

Possible server crash

These functions might create a server crash

Warning

Experimental functions

They are not officially of the current release.
They likely will not be officially be part of the next release:
- The functions might not make use of ANY-INTEGER and ANY-NUMERICAL
- Name might change.
- Signature might change.
- Functionality might change.
- pgTap tests might be missing.
- Might need c/c++ coding.
- May lack documentation.
- Documentation if any might need to be rewritten.
- Documentation examples might need to be automatically generated.
- Might need a lot of feedback from the comunity.
- Might depend on a proposed function of pgRouting
- Might depend on a deprecated function of pgRouting

Availability

Version 3.0.0
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

TODO check which statement is true:
- The cost value of all input edges must be nonnegative.
- Process is done when the cost value of all input edges is nonnegative.
- Process is done on edges with nonnegative cost.
Running time: \(O(U * (E + V * logV))\)
- where \(U\) is the value of the max flow.
- \(U\) is upper bound on number of iterations. In many real world cases number of iterations is much smaller than \(U\).

Signatures¶

Summary

pgr_maxFlowMinCost(Edges SQL, source,  target)
pgr_maxFlowMinCost(Edges SQL, sources, target)
pgr_maxFlowMinCost(Edges SQL, source,  targets)
pgr_maxFlowMinCost(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

One to One¶

pgr_maxFlowMinCost(Edges SQL, source, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

Example:	From vertex \(2\) to vertex \(3\)

SELECT * FROM pgr_MaxFlowMinCost(
    'SELECT id,
     source, target,
     capacity, reverse_capacity,
     cost, reverse_cost FROM edge_table',
    2, 3
);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    4 |      2 |      5 |   80 |                20 |   80 |       80
   2 |    3 |      4 |      3 |   80 |                50 |   80 |      160
   3 |    8 |      5 |      6 |   80 |                20 |   80 |      240
   4 |    9 |      6 |      9 |   80 |                50 |   80 |      320
   5 |   16 |      9 |      4 |   80 |                 0 |   80 |      400
(5 rows)

One to Many¶

pgr_maxFlowMinCost(Edges SQL, source, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

Example:	From vertex \(13\) to vertices \(\{7, 1, 4\}\)

SELECT * FROM pgr_MaxFlowMinCost(
    'SELECT id,
     source, target,
     capacity, reverse_capacity,
     cost, reverse_cost FROM edge_table',
    13, ARRAY[7, 1, 4]
);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    1 |      2 |      1 |   50 |                80 |   50 |       50
   2 |    4 |      5 |      2 |   50 |                 0 |   50 |      100
   3 |   16 |      9 |      4 |   50 |                30 |   50 |      150
   4 |   10 |     10 |      5 |   50 |                 0 |   50 |      200
   5 |   12 |     10 |     11 |   50 |                50 |   50 |      250
   6 |   13 |     11 |     12 |   50 |                50 |   50 |      300
   7 |   15 |     12 |      9 |   50 |                 0 |   50 |      350
   8 |   14 |     13 |     10 |  100 |                30 |  100 |      450
(8 rows)

Many to One¶

pgr_maxFlowMinCost(Edges SQL, sources, target)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

Example:	From vertices \(\{1, 7, 14\}\) to vertex \(12\)

SELECT * FROM pgr_MaxFlowMinCost(
    'SELECT id,
     source, target,
     capacity, reverse_capacity,
     cost, reverse_cost FROM edge_table',
    ARRAY[1, 7, 14], 12
);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    1 |      1 |      2 |   80 |                 0 |   80 |       80
   2 |    4 |      2 |      5 |   80 |                20 |   80 |      160
   3 |    8 |      5 |      6 |  100 |                 0 |  100 |      260
   4 |   10 |      5 |     10 |   30 |               100 |   30 |      290
   5 |    9 |      6 |      9 |   50 |                80 |   50 |      340
   6 |   11 |      6 |     11 |   50 |                80 |   50 |      390
   7 |    6 |      7 |      8 |   50 |                 0 |   50 |      440
   8 |    7 |      8 |      5 |   50 |                 0 |   50 |      490
   9 |   15 |      9 |     12 |   50 |                30 |   50 |      540
  10 |   12 |     10 |     11 |   30 |                70 |   30 |      570
  11 |   13 |     11 |     12 |   80 |                20 |   80 |      650
(11 rows)

Many to Many¶

pgr_maxFlowMinCost(Edges SQL, sources, targets)
RETURNS SET OF (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

Example:	From vertices \(\{7, 13\}\) to vertices \(\{3, 9\}\)

SELECT * FROM pgr_MaxFlowMinCost(
    'SELECT id,
     source, target,
     capacity, reverse_capacity,
     cost, reverse_cost FROM edge_table',
    ARRAY[7, 13], ARRAY[3, 9]
);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    8 |      5 |      6 |  100 |                 0 |  100 |      100
   2 |    9 |      6 |      9 |  100 |                30 |  100 |      200
   3 |    6 |      7 |      8 |   50 |                 0 |   50 |      250
   4 |    7 |      8 |      5 |   50 |                 0 |   50 |      300
   5 |   10 |     10 |      5 |   50 |                 0 |   50 |      350
   6 |   12 |     10 |     11 |   50 |                50 |   50 |      400
   7 |   13 |     11 |     12 |   50 |                50 |   50 |      450
   8 |   15 |     12 |      9 |   50 |                 0 |   50 |      500
   9 |   14 |     13 |     10 |  100 |                30 |  100 |      600
(9 rows)

Parameters¶

Column	Type	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`		Capacity 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	Capacity of the edge (target, source), When negative: edge (target, source) does not exist, therefore it’s not part of the graph.
cost	`ANY-NUMERICAL`		Weight of the edge (source, target) if it exists.
reverse_cost	`ANY-NUMERICAL`	0	Weight of the edge (target, source) if it exists.

Where:

ANY-INTEGER:	SMALLINT, INTEGER, BIGINT
ANY-NUMERICAL:	smallint, int, bigint, real, float

Result Columns¶

Column	Type	Description
seq	`INT`	Sequential value starting from 1.
edge	`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).
cost	`FLOAT`	The cost of sending this flow through the edge in the direction (source, target).
agg_cost	`FLOAT`	The aggregate cost.

pgr_maxFlowMinCost - Experimental¶

Description¶

Signatures¶

One to One¶

One to Many¶

Many to One¶

Many to Many¶

Parameters¶

Inner query¶

Result Columns¶

See Also¶