Ford-Fulkerson
Repeatedly find any augmenting path from s to t in residual graph. Push flow along it. Terminate when no path.
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Residual graph
For each edge (u,v) with capacity c and flow f: forward residual c-f, backward residual f. Path uses residual capacities.
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Max-flow min-cut
Max flow value = min s-t cut capacity. Fundamental duality.