Warning
Experimental functions
Characteristics
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.
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 the 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:
Maximum flow problem
Given:
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: