maximum-flow

To model a max-flow problem with multiple sources and sinks, you add a super-source connected to all sources and a super-sink connected from all sinks, applying infinite capacities to these new edges while maintaining standard network conservation constraints.

Choose Edmonds-Karp over Ford-Fulkerson when you need guaranteed polynomial time complexity, as Edmonds-Karp uses breadth-first search for augmenting paths, preventing the potential infinite loops of standard Ford-Fulkerson on irrational capacities.

You extract a min-cut certificate by finding all vertices reachable from the source in the residual network, defining the cut partition, and proving optimality by demonstrating that the flow equals the capacity across this specific cut.

Yes, you can solve bipartite matching by transforming it into a max-flow model, connecting a source to one partition and the other to a sink with unit capacities, so the maximum flow equals the maximum matching.

Scaling algorithms for maximum flow require additional complexity to manage capacity thresholds and may be unnecessary for simple networks, though they prevent worst-case augmenting path behaviors found in naive Ford-Fulkerson implementations.

You verify maximum flow optimality by constructing the residual network, identifying a saturated min-cut, and providing a mathematical certificate proving no augmenting paths exist from the source to the sink.