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Unformatted text preview: C H A P T E R 4 THE PRIMAL-DUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS Michel X. Goemans David P. Williamson Dedicated to the memory of Albert W. Tucker The primal-dual method is a standard tool in the de- sign of algorithms for combinatorial optimization problems. This chapter shows how the primal-dual method can be modified to provide good approximation algorithms for a wide variety of NP-hard problems. We concentrate on re- sults from recent research applying the primal-dual method to problems in network design. INTRODUCTION 4.1 In the last four decades, combinatorial optimization has been strongly influenced by linear programming. With the mathematical and algorithmic understanding of linear programs came a whole host of ideas and tools that were then applied to combinatorial optimization. Many of these ideas and tools are still in use today, and form the bedrock of our understanding of combinatorial optimization. Oneofthesetoolsis the primal-dualmethod .It wasproposedbyDantzig,Ford,and Fulkerson[DFF56]asanothermeansofsolvinglinearprograms.Ironically,theirinspira- tioncamefromcombinatorialoptimization.Intheearly1930s,Egervary[Ege31]proved 144 4.1 INTRODUCTION 145 a min-max relation for the assignment problem (or the minimum-cost bipartite perfect matching problem) by reducing it to a known min-max result for maximum cardinality matchings. This lead Kuhn to propose his primal-dual Hungarian Method for solving theassignmentproblem[Kuh55],whichtheninspiredDantzig,Ford,andFulkerson.Al- though the primal-dual method in its original form has not survived as an algorithm for linear programming,it has found widespread use as a means of devising algorithms for problems in combinatorial optimization. The main feature of the primal-dual method is that it allows a weighted optimization problem to be reduced to a purely combinatorial, unweightedproblem.Mostofthefundamentalalgorithmsincombinatorialoptimization either use this method or can be understood in terms of it, including Dijkstras shortest pathalgorithm[Dij59],FordandFulkersonsnetworkflowalgorithm[FF56],Edmonds non-bipartitematchingalgorithm[Edm65]and,ofcourse,Kuhnsassignmentalgorithm. The primal-dual method as described above has been used to solve problems that can be modelled as linear programs; the method simply leads to efficient polynomial- timealgorithmsforsolvingtheseproblems.Since NP-hardproblemscannotbemodelled as polynomially-sized linear programs unless P = NP , the primal-dual method does not generalize straightforwardly to generate algorithms for the NP-hard optimization problems that are the interest of this book. Nevertheless, with modifications the primal- dual methodleads to approximationalgorithms fora wide varietyof NP-hardproblems....
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This note was uploaded on 02/19/2012 for the course MATH 482 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08