The Ford-Fulkerson algorithm repeatedly augments a feasible flow along an augmenting path. When no such path remains, the source-reachable vertices in the residual network define a cut of capacity equal to the flow value, proving optimality by the max-flow min-cut theorem.
Starting from the zero flow with integer capacities, every residual capacity and every augmenting bottleneck is an integer. Each augmentation therefore preserves integer edge flows and raises the flow value by at least one. Since the value is bounded by the total capacity leaving the source, the algorithm terminates after finitely many augmentations with an integral maximum flow.
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The Ford–Fulkerson algorithm is a method used to compute the maximum flow in a flow network. Developed by L.R. Ford, Jr. and D.R. Fulkerson in the 1950s, this algorithm is based on the concept of augmenting paths and works by iteratively increasing the flow in the network until no more augmenting paths can be found.