How much can you push through a network of pipes, roads, or wires? The max-flow problem finds the answer — and reveals the bottleneck that limits it.
Maximum FlowLive
Edmonds-Karp network flow
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The max-flow problem asks how much can be pushed from a source to a sink through a network of capacity-limited edges. Edmonds-Karp repeatedly finds an augmenting path and saturates it until none remain. By the max-flow min-cut theorem, the answer equals the capacity of the cheapest set of edges that, if cut, disconnects source from sink — the bottleneck.
Reading this result: Both feed edges are generous, so the bottleneck has moved to the fixed interior capacities — adding more feed capacity no longer raises the max flow (min-cut theorem).
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How it works
Given a network of edges with capacities, the maximum-flow problem finds the greatest total flow from source to sink. The Edmonds-Karp algorithm repeatedly finds a shortest augmenting path and saturates it. By the celebrated max-flow min-cut theorem, the maximum flow equals the capacity of the smallest set of edges whose removal disconnects source from sink — the true bottleneck.
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