# 26_3 - Contents Maximum flow problem Minimum cut problem...

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1 Contents Maximum flow problem. Minimum cut problem. Max-flow min-cut theorem. Augmenting path algorithm. Capacity-scaling. Shortest augmenting path. Max-Flow Min-Cut Theorem MAX-FLOW MIN-CUT THEOREM (Ford-Fulkerson, 1956): In any network, the value of the max flow is equal to the value of the min cut. Proof: ? Flow value = 28 s 2 3 4 5 6 7 t 15 5 30 15 10 8 15 9 6 10 10 10 15 4 4 10 9 9 14 14 4 10 4 8 9 1 0 0 0 0 Cut capacity = 28 Flow value = 0 s 4 2 5 3 t 4 0 0 0 0 0 0 4 0 4 4 10 13 10 Towards an Algorithm Find an s-t path where each edge e=(u,v) has c(e) > f(e) and "augment" flow along the path. Towards an Algorithm Find an s-t path where each arc e=(u,v) has c(e) > f(e) and "augment" flow along the path. Repeat until you get stuck. Flow value = 10 s 4 2 5 3 t 4 0 0 0 0 0 0 4 0 4 4 10 13 10 10 10 10 The design of this algorithm? Towards an Algorithm Find an s-t path where each arc has u(e) > f(e) and "augment" flow along the path. Repeat until you get stuck. The straight forward greedy algorithm fails. Flow value = 10 s 4 2 5 3 t 10 13 10 4 0 0 0 4 0 4 4 10 10 10 Flow value = 14 s 4 2 5 3 t 10 13 10 4 4 4 4 10 4 4 10 6 4 4 Residual Network Let f be a flow on G = (V, E). The residual network G f = (V, E f )

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