push_relabel_max_flow

graph_tool.flow.push_relabel_max_flow(g, source, target, capacity, residual=None)

Calculate maximum flow on the graph with the push-relabel algorithm.

Parameters:

gGraph

Graph to be used.

sourceVertex

The source vertex.

targetVertex

The target (or “sink”) vertex.

capacityEdgePropertyMap

Edge property map with the edge capacities.

residualEdgePropertyMap (optional, default: none)

Edge property map where the residuals should be stored.

Returns:

residualEdgePropertyMap

Edge property map with the residual capacities (capacity - flow).

Notes

The algorithm is defined in [goldberg-new-1985]. The complexity is \(O(V^3)\).

References

[boost-push-relabel]

http://www.boost.org/libs/graph/doc/push_relabel_max_flow.html

[goldberg-new-1985]

A. V. Goldberg, “A New Max-Flow Algorithm”, MIT Tehnical report MIT/LCS/TM-291, 1985.

Examples

>>> g = gt.load_graph("flow-example.xml.gz")
>>> cap = g.edge_properties["cap"]
>>> src, tgt = g.vertex(0), g.vertex(1)
>>> res = gt.push_relabel_max_flow(g, src, tgt, cap)
>>> res.a = cap.a - res.a  # the actual flow
>>> max_flow = sum(res[e] for e in tgt.in_edges())
>>> print(max_flow)
44.8905957841...
>>> pos = g.vertex_properties["pos"]
>>> gt.graph_draw(g, pos=pos, edge_pen_width=gt.prop_to_size(res, mi=0, ma=5, power=1), output="example-push-relabel.pdf")
<...>

Edge flows obtained with the push-relabel algorithm. The source and target are on the upper left and lower right corners, respectively. The edge flows are represented by the edge width.