# boykov_kolmogorov_max_flow#

graph_tool.flow.boykov_kolmogorov_max_flow(g, source, target, capacity, residual=None)[source]#

Calculate maximum flow on the graph with the Boykov-Kolmogorov 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 [kolmogorov-graph-2003] and [boykov-experimental-2004]. The worst case complexity is $$O(EV^2|C|)$$, where $$|C|$$ is the minimum cut (but typically performs much better).

For a more detailed description, see [boost-kolmogorov].

References

Vladimir Kolmogorov, “Graph Based Algorithms for Scene Reconstruction from Two or More Views”, PhD thesis, Cornell University, September 2003.

Yuri Boykov and Vladimir Kolmogorov, “An Experimental Comparison of Min-Cut/Max-Flow Algorithms for Energy Minimization in Vision”, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1124-1137, Sept. 2004. DOI: 10.1109/TPAMI.2004.60 [sci-hub, @tor]

Examples

>>> g = gt.load_graph("flow-example.xml.gz")
>>> cap = g.edge_properties["cap"]
>>> src, tgt = g.vertex(0), g.vertex(1)
>>> res = gt.boykov_kolmogorov_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=3, power=1), output="example-kolmogorov.pdf")
<...>