Design of Approximation Algorithms

304 Further uses of deterministic rounding of linear programsbetween i and j for all i, j ∈ V . In Section 16.4, we show that this variant of the survivablenetwork design problem is substantially harder to approximate that the edge-disjoint versionwe have considered above.Exercises11.1 We consider a variant of the generalized assignment problem without costs. Suppose weare given a set of n jobs to be assigned to m machines. Each job j is to be scheduledon exactly one machine. If job j is scheduled on machine i, then it requires p ij units ofprocessing time. The goal is to find a schedule of minimum length, which is equivalent tofinding an assignment of jobs to machines that minimizes the maximum total processingtime required by a machine. This problem is sometimes called scheduling unrelated parallelmachines so as to minimize the makespan. We show that deterministic rounding of a linearprogram can be used to develop a polynomial-time 2-relaxed decision procedure (recallthe definition of a relaxed decision procedure from Exercise 2.4).Consider the following set of linear inequalities for a parameter T :m∑x ij = 1, j = 1, . . . , n,i=1n∑p ij x ij ≤ T, i = 1, . . . , m,j=1x ij ≥ 0,x ij = 0, if p ij > T.i = 1, . . . , m, j = 1, . . . , nIf a feasible solution exists, let x be a basic feasible solution for this set of linear inequalities.Consider the bipartite graph G on machine nodes M 1 , . . . , M m and job nodesN 1 , . . . , N n with edges (M i , N j ) for each variable x ij > 0.(a) Prove that the linear inequalities are a relaxation of the problem, in the sense that ifthe length of the optimal schedule is T , then there is a feasible solution to the linearinequalities.(b) Prove that each connected component of G of k nodes has exactly k edges, and so isa tree plus one additional edge.(c) If x ij = 1, assign job j to machine i. Once all of these jobs are assigned, use thestructure of the previous part to show that it is possible to assign at most oneadditional job to any machine. Argue that this results in a schedule of length atmost 2T .(d) Use the previous parts to give a polynomial-time 2-relaxed decision procedure, andconclude that there is a polynomial-time 2-approximation algorithm for schedulingunrelated parallel machines to minimize the makespan.11.2 In this exercise, we prove Theorem 11.27.(a) First, prove the following. Given two tight sets A and B, one of the following twostatements must hold:• A ∪ B and A ∩ B are tight, and χ δ(A) + χ δ(B) = χ δ(A∩B) + χ δ(A∪B) ; orElectronic web edition. Copyright 2011 by David P. Williamson and David B. Shmoys.To be published by Cambridge University Press