hw7520_11_3 - CS 7250, Approximation Algorithms Homework 3...

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Unformatted text preview: 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 Vaziranis 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 , ( 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 , i V (2) f ij , ( i,j ) E Prove that the constraints in (2): j :( j,i ) E f ji- j :( i,j ) E f ij , i V , imply the flow preser-...
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This note was uploaded on 10/23/2011 for the course CS 7520 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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hw7520_11_3 - CS 7250, Approximation Algorithms Homework 3...

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