Sol8 - AMS/MBA 546(Spring 2010 Estie Arkin Network Flows Solution sketch to homework set 8 1[9.30 Use case analysis to show that if x ∗ satisfies

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Unformatted text preview: AMS/MBA 546 (Spring, 2010) Estie Arkin Network Flows: Solution sketch to homework set # 8 1). [9.30] Use case analysis to show that if x ∗ satisfies complementary slackness conditions from theorem 9.4 with respect to node potentials π and arc capacities u , then it must also satisfy these optimality conditions with node potentials π and capacities u ′ . Note that c π ij remains the same. Also, when x ∗ ij = u ij , we have u ij = u ′ ij , and thus (9.8c) is the same. If x ∗ ij < u ij then u ′ ij = ∞ , but this has no effect on (9.8a) and (9.8b), so they still hold. Another approach to proving the claim is to show that the residual graphs with capacities u and u ′ contain exactly the same arcs, and have the same costs. They only differ in the capacity of some arcs. Why? If x ∗ ij > 0 an arc ( j,i ) with r ji = x ∗ ij and cost- c ij exists in the residual graph both for the original and modified capacities. If x ∗ ij < u ij then a forward arc ( i,j ) with cost c ij...
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This note was uploaded on 01/31/2011 for the course AMS 546 taught by Professor Arkin,e during the Fall '08 term at SUNY Stony Brook.

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