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**Unformatted text preview: **Com S 511: Homework #1 Due on Friday, September 4, 2009 1 Com S 511 : Homework #1 Problem 1 Problem 1 Because the network has no infinite-capacity paths from s to t. So along each path from s to t, we could not send a finite among of flow, thus the maximum folw f * is finite. By (7.13), the capacity of min cut is equal to the value of the max flow. Let (A,B) denotes such cut and E c denotes the set of edges crossing from A to B. Then any edge crossing from A to B has finite capacity, which means E c E . Thus c ( A,B ) = e out of A c e e E c e . Let any cut (A, B) be any cut in the new network G. If 1) among the edges crossing from A to B, there exists an edge, which has infinite capacity in G, then c(A, B) M e E c e c(A, B); otherwise 2), the capacity of (A, B) remains unchanged and c(A, B) c(A,B). Therefore, (A, B) is still the min cut in G and thus the max flow is not affected....

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