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 ANYINTEGER and ANYNUMERICAL
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 

Edges query as described in Inner Queries. 

Combinations SQL 

Combinations query as described in Inner Queries. 

source 

Identifier of the starting vertex of the flow. 

sources 

Array of identifiers of the starting vertices of the flow. 

target 

Identifier of the ending vertex of the flow. 

targets 

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 

Identifier of the edge. 

source 

Identifier of the first end point vertex of the edge. 

target 

Identifier of the second end point vertex of the edge. 

capacity 

Weight of the edge (source, target)


reverse_capacity 

1 
Weight of the edge (target, source),

Where:
 ANYINTEGER
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 

Identifier of the edge. 

source 

Identifier of the first end point vertex of the edge. 

target 

Identifier of the second end point vertex of the edge. 

capacity 

Capacity of the edge (source, target)


reverse_capacity 

1 
Capacity of the edge (target, source),

cost 

Weight of the edge (source, target) if it exists. 

reverse_cost 

0 
Weight of the edge (target, source) if it exists. 
Where:
 ANYINTEGER
SMALLINT, INTEGER, BIGINT
 ANYNUMERICAL
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 

Identifier of the first end point vertex of the edge. 

target 

Identifier of the second end point vertex of the edge. 
Where:
 ANYINTEGER
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 

Sequential value starting from 1. 
edge 

Identifier of the edge in the original query(edges_sql). 
start_vid 

Identifier of the first end point vertex of the edge. 
end_vid 

Identifier of the second end point vertex of the edge. 
flow 

Flow through the edge in the direction ( 
residual_capacity 

Residual capacity of the edge in the direction ( 
For pgr_maxFlowMinCost  Experimental
Column 
Type 
Description 

seq 

Sequential value starting from 1. 
edge 

Identifier of the edge in the original query(edges_sql). 
source 

Identifier of the first end point vertex of the edge. 
target 

Identifier of the second end point vertex of the edge. 
flow 

Flow through the edge in the direction (source, target). 
residual_capacity 

Residual capacity of the edge in the direction (source, target). 
cost 

The cost of sending this flow through the edge in the direction (source, target). 
agg_cost 

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¶
Indices and tables