hw1_sol_long - Com S 511 Homework#1 Due on Friday September...

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Com S 511: Homework #1 Due on Friday, September 4, 2009 1
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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 0 . Thus c ( A, B ) = e out of A c e e E 0 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 0 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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