# pgRouting Manual (3.3)

## Flow - Family of functions

- Supported versions: Latest (3.3) 3.2 3.1 3.0

- Unsupported versions: 2.6 2.5 2.4 2.3

# Flow - Family of functions

- pgr_maxFlow - Only the Max flow calculation using Push and Relabel algorithm.

- pgr_boykovKolmogorov - Boykov and Kolmogorov with details of flow on edges.

- pgr_edmondsKarp - Edmonds and Karp algorithm with details of flow on edges.

- pgr_pushRelabel - Push and relabel algorithm with details of flow on edges.

- Applications

  - pgr_edgeDisjointPaths - Calculates edge disjoint paths between two groups of vertices.

  - pgr_maxCardinalityMatch - Calculates a maximum cardinality matching in a graph.

Experimental

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

- pgr_maxFlowMinCost - Experimental - Details of flow and cost on edges.

- pgr_maxFlowMinCost_Cost - Experimental - Only the Min Cost calculation.

## Flow Functions General Information

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

pgr_maxFlow is the maximum Flow and that maximum is guaranteed to be the same on the functions pgr_pushRelabel, pgr_edmondsKarp, pgr_boykovKolmogorov, but the actual flow through each edge may vary.

## 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

For pgr_pushRelabel, pgr_edmondsKarp, pgr_boykovKolmogorov :

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

For pgr_maxFlowMinCost - Experimental and pgr_maxFlowMinCost_Cost - Experimental:

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

For pgr_pushRelabel, pgr_edmondsKarp, pgr_boykovKolmogorov, pgr_edgeDisjointPaths, pgr_maxFlowMinCost and pgr_maxFlowMinCost_Cost :

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

For pgr_pushRelabel, pgr_edmondsKarp, pgr_boykovKolmogorov :

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`).

For pgr_maxFlowMinCost - Experimental

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.

## Adcanced Documentation

A flow network is a directed graph where each edge has a capacity and a flow. The flow through an edge must not exceed the capacity of the edge. Additionally, the incoming and outgoing flow of a node must be equal except for source which only has outgoing flow, and the destination(sink) which only has incoming flow.

Maximum flow algorithms calculate the maximum flow through the graph and the flow of each edge.

The maximum flow through the graph is guaranteed to be the same with all implementations, but the actual flow through each edge may vary. Given the following query:

pgr_maxFlow \((edges\_sql, source\_vertex, sink\_vertex)\)

where \(edges\_sql = \{(id_i, source_i, target_i, capacity_i, reverse\_capacity_i)\}\)

Graph definition

The weighted directed graph, \(G(V,E)\), is defined as:

- the set of vertices \(V\)

  - \(source\_vertex \cup sink\_vertex \bigcup source_i \bigcup target_i\)

- the set of edges \(E\)

  - \(E = \begin{cases} \text{ } \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \quad \text{ if } reverse\_capacity = \varnothing \\ \text{ } & \quad \text{ } \\ \{(source_i, target_i, capacity_i) \text{ when } capacity > 0 \} & \text{ } \\ \cup \{(target_i, source_i, reverse\_capacity_i) \text{ when } reverse\_capacity_i > 0)\} & \quad \text{ if } reverse\_capacity \neq \varnothing \\ \end{cases}\)

Maximum flow problem

Given:

- \(G(V,E)\)

- \(source\_vertex \in V\) the source vertex

- \(sink\_vertex \in V\) the sink vertex

Then:

- \(pgr\_maxFlow(edges\_sql, source, sink) = \boldsymbol{\Phi}\)

- \(\boldsymbol{\Phi} = {(id_i, edge\_id_i, source_i, target_i, flow_i, residual\_capacity_i)}\)

Where:

\(\boldsymbol{\Phi}\) is a subset of the original edges with their residual capacity and flow. The maximum flow through the graph can be obtained by aggregating on the source or sink and summing the flow from/to it. In particular:

- \(id_i = i\)

- \(edge\_id = id_i\) in edges_sql

- \(residual\_capacity_i = capacity_i - flow_i\)

## See Also

- https://en.wikipedia.org/wiki/Maximum_flow_problem

Indices and tables

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