# pgRouting Manual (3.3)

## pgr_maxFlowMinCost - Experimental

- Supported versions: Latest (3.3) 3.2 3.1 3.0

# 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 3.2.0

  - New experimental function:

    - pgr_maxFlowMinCost(Combinations)

- 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, source,  target)
    pgr_maxFlowMinCost(Edges SQL, sources, target)
    pgr_maxFlowMinCost(Edges SQL, source,  targets)
    pgr_maxFlowMinCost(Edges SQL, sources, targets)
    pgr_maxFlowMinCost(Edges SQL, Combinations SQL)
    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)

### Combinations

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

Example

  Using a combinations table, equivalent to calculating result 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',
        'SELECT * FROM ( VALUES (7, 3), (13, 9) ) AS t(source, target)'
    );
     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 | 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`

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

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).
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.

## See Also

- Flow - Family of functions

- https://www.boost.org/libs/graph/doc/successive_shortest_path_nonnegative_weights.html

Indices and tables

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