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hw7520_11_3

hw7520_11_3 - CS 7250 Approximation Algorithms Homework 3...

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CS 7250, Approximation Algorithms Homework 3 Thu, Feb 17, 2011 Due Thu, Feb 24, 2011 Problem 1: Primal-Dual, Exact Complementary Slackness This exercise is a review of the basics of LP-duality (chaper 12 in Vazirani’s book). (a) Let G ( V, E ) be an undirected graph with source s V and sink t V \{ s } , and edge capacities c : E R + . Replace each undirected edge { i, j } with two directed edges ( i, j ) and ( j, i ) of equal capacities c ij = c ji . Introduce a link of infinite capacity from t to s , and a link of zero capacity from s to t : c ts = and c st =0. Then the straightforward formalization of max s - t flow in G is: maximize f ts subject to f ij c ij , ( i, j ) E j :( j,i ) E f ji - j :( i,j ) E f ij = 0 , i V (1) f ij 0 , ( i, j ) E In class we formalized max s - t flow with the slightly different LP (2) below, where the set of flow preservation constraints of (1) are replaced with a set of corresponding constraints that may initially seem weaker. maximize f ts subject to f ij c ij , ( i, j ) E j :( j,i ) E f ji - j :( i,j ) E f ij 0 , i V (2) f ij 0 , ( i, j ) E Prove that the constraints in (2): j :( j,i ) E f ji - j :( i,j ) E f ij 0 , i V , imply the flow preser- vation constraints in (1): j :( j,i ) E f ji - j :( i,j ) E f

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hw7520_11_3 - CS 7250 Approximation Algorithms Homework 3...

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