pgr_maxFlowMinCost - Experimental

pgr_maxFlowMinCost — Calculates the edges that minimizes the total cost of the maximum flow on a graph

_images/boost-inside.jpeg

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 3.2.0

    • New experimental function:

  • Version 3.0.0

  • 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

  • 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, start vid, end vid)
pgr_maxFlowMinCost(Edges SQL, start vid, end vids)
pgr_maxFlowMinCost(Edges SQL, start vids, end vid)
pgr_maxFlowMinCost(Edges SQL, start vids, end vids)
pgr_maxFlowMinCost(Edges SQL, Combinations SQL)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET

One to One

pgr_maxFlowMinCost(Edges SQL, start vid, end vid)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
Example:

From vertex \(11\) to vertex \(12\)

SELECT * FROM pgr_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  11, 12);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |   10 |      7 |      8 |  100 |                30 |  100 |      100
   2 |   12 |      8 |     12 |  100 |                 0 |  100 |      200
   3 |    8 |     11 |      7 |  100 |                30 |  100 |      300
   4 |   11 |     11 |     12 |  130 |                 0 |  130 |      430
(4 rows)

One to Many

pgr_maxFlowMinCost(Edges SQL, start vid, end vids)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
Example:

From vertex \(11\) to vertices \(\{5, 10, 12\}\)

SELECT * FROM pgr_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  11, ARRAY[5, 10, 12]);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    1 |      6 |      5 |   30 |               100 |   30 |       30
   2 |    4 |      7 |      6 |   30 |                20 |   30 |       60
   3 |   10 |      7 |      8 |  100 |                30 |  100 |      160
   4 |   12 |      8 |     12 |  100 |                 0 |  100 |      260
   5 |    8 |     11 |      7 |  130 |                 0 |  130 |      390
   6 |   11 |     11 |     12 |  130 |                 0 |  130 |      520
   7 |    9 |     11 |     16 |   80 |                50 |   80 |      600
   8 |    3 |     15 |     10 |   80 |                50 |   80 |      680
   9 |   16 |     16 |     15 |   80 |                 0 |   80 |      760
(9 rows)

Many to One

pgr_maxFlowMinCost(Edges SQL, start vids, end vid)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
Example:

From vertices \(\{11, 3, 17\}\) to vertex \(12\)

SELECT * FROM pgr_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  ARRAY[11, 3, 17], 12);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    7 |      3 |      7 |   50 |                 0 |   50 |       50
   2 |   10 |      7 |      8 |  100 |                30 |  100 |      150
   3 |   12 |      8 |     12 |  100 |                 0 |  100 |      250
   4 |    8 |     11 |      7 |   50 |                80 |   50 |      300
   5 |   11 |     11 |     12 |  130 |                 0 |  130 |      430
(5 rows)

Many to Many

pgr_maxFlowMinCost(Edges SQL, start vids, end vids)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
Example:

From vertices \(\{11, 3, 17\}\) to vertices \(\{5, 10, 12\}\)

SELECT * FROM pgr_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  ARRAY[11, 3, 17], ARRAY[5, 10, 12]);
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    7 |      3 |      7 |   50 |                 0 |   50 |       50
   2 |    1 |      6 |      5 |   50 |                80 |   50 |      100
   3 |    4 |      7 |      6 |   50 |                 0 |   50 |      150
   4 |   10 |      7 |      8 |  100 |                30 |  100 |      250
   5 |   12 |      8 |     12 |  100 |                 0 |  100 |      350
   6 |    8 |     11 |      7 |  100 |                30 |  100 |      450
   7 |   11 |     11 |     12 |  130 |                 0 |  130 |      580
   8 |    9 |     11 |     16 |   30 |               100 |   30 |      610
   9 |    3 |     15 |     10 |   80 |                50 |   80 |      690
  10 |   16 |     16 |     15 |   80 |                 0 |   80 |      770
  11 |   15 |     17 |     16 |   50 |                 0 |   50 |      820
(11 rows)

Combinations

pgr_maxFlowMinCost(Edges SQL, Combinations SQL)
RETURNS (seq, edge, source, target, flow, residual_capacity, cost, agg_cost)
OR EMPTY SET
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_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  'SELECT * FROM combinations WHERE target NOT IN (5, 6)');
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    4 |      6 |      7 |   80 |                20 |   80 |       80
   2 |    8 |      7 |     11 |   80 |                20 |   80 |      160
   3 |    9 |     11 |     16 |   80 |                50 |   80 |      240
   4 |   16 |     16 |     15 |   80 |                 0 |   80 |      320
(4 rows)

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

Capacity of the edge (source, target)

  • When negative: edge (target, source) 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 exist

reverse_cost

ANY-NUMERICAL

\(-1\)

Weight of the edge (target, source) if it exist

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

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.

Additional Examples

Example:

Manually assigned vertex combinations.

SELECT * FROM pgr_maxFlowMinCost(
  'SELECT id, source, target, capacity, reverse_capacity, cost, reverse_cost
  FROM edges',
  'SELECT * FROM (VALUES (5, 10), (6, 15), (6, 14)) AS t(source, target)');
 seq | edge | source | target | flow | residual_capacity | cost | agg_cost
-----+------+--------+--------+------+-------------------+------+----------
   1 |    4 |      6 |      7 |   80 |                20 |   80 |       80
   2 |    8 |      7 |     11 |   80 |                20 |   80 |      160
   3 |    9 |     11 |     16 |   80 |                50 |   80 |      240
   4 |   16 |     16 |     15 |   80 |                 0 |   80 |      320
(4 rows)

See Also

Indices and tables