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Unformatted text preview: C H A P T E R 4 THE PRIMALDUAL 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 primaldual method is a standard tool in the de sign of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on re sults from recent research applying the primaldual 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 primaldualmethod .It wasproposedbyDantzig,Ford,and Fulkerson[DFF56]asanothermeansofsolvinglinearprograms.Ironically,theirinspira tioncamefromcombinatorialoptimization.Intheearly1930s,Egerv´ary[Ege31]proved 144 4.1 INTRODUCTION 145 a minmax relation for the assignment problem (or the minimumcost bipartite perfect matching problem) by reducing it to a known minmax result for maximum cardinality matchings. This lead Kuhn to propose his primaldual “Hungarian Method” for solving theassignmentproblem[Kuh55],whichtheninspiredDantzig,Ford,andFulkerson.Al though the primaldual 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 primaldual 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 Dijkstra’s shortest pathalgorithm[Dij59],FordandFulkerson’snetworkflowalgorithm[FF56],Edmonds’ nonbipartitematchingalgorithm[Edm65]and,ofcourse,Kuhn’sassignmentalgorithm. The primaldual 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 NPhardproblemscannotbemodelled as polynomiallysized linear programs unless P = NP , the primaldual method does not generalize straightforwardly to generate algorithms for the NPhard optimization problems that are the interest of this book. Nevertheless, with modifications the primal dual methodleads to approximationalgorithms fora wide varietyof NPhardproblems....
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 Spring '08
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 Linear Programming, Approximation, Optimization, Linear Programming Relaxation, Computational problems in graph theory, Approximation algorithm, primaldual method

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