By Vijay V. Vazirani

ISBN-10: 3642084699

ISBN-13: 9783642084690

ISBN-10: 3662045656

ISBN-13: 9783662045657

This ebook covers the dominant theoretical methods to the approximate answer of difficult combinatorial optimization and enumeration difficulties. It comprises based combinatorial concept, helpful and engaging algorithms, and deep effects concerning the intrinsic complexity of combinatorial difficulties. Its readability of exposition and ideal choice of workouts will make it available and attractive to all people with a flavor for arithmetic and algorithms.

Richard Karp,University Professor, collage of California at Berkeley

Following the improvement of uncomplicated combinatorial optimization thoughts within the Nineteen Sixties and Seventies, a first-rate open query used to be to improve a conception of approximation algorithms. within the Nineties, parallel advancements in concepts for designing approximation algorithms in addition to tools for proving hardness of approximation effects have resulted in a gorgeous idea. the necessity to resolve really huge circumstances of computationally challenging difficulties, akin to these coming up from the web or the human genome venture, has additionally elevated curiosity during this idea. the sector is at the moment very lively, with the toolbox of approximation set of rules layout concepts getting constantly richer.

It is a excitement to suggest Vijay Vazirani's well-written and finished ebook in this vital and well timed subject. i'm convinced the reader will locate it most precious either as an creation to approximability in addition to a connection with the various features of approximation algorithms.

László Lovász, Senior Researcher, Microsoft Research

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**Additional resources for Approximation Algorithms**

**Sample text**

V. 3 (Multiway cut) 1. For each i = 1, ... , k, compute a minimum weight isolating cut for si. say Ci. 2. Discard the heaviest of these cuts, and output the union of the rest, say C. Each computation in Step 1 can be accomplished by identifying the terminals in S- {si} into a single node, and finding a minimum cut separating this node from si; this takes one max-flow computation. Clearly, removing C from the graph disconnects every pair of terminals, and so is a multiway cut. 3 achieves an approximation guarantee of2-2jk.

The process is schematically shown in the following figure. 0] Gl Let t 0 , ... , tk- 1 be the degree-weighted functions defined on graphs Go, ... , Gk_ 1 . The vertex cover chosen is C = WoU ... UWk-1· Clearly, V -C = DoU .. UDk. Theorem 2. 7 The layer algorithm achieves an approximation guarantee of factor 2 for the vertex cover problem, assuming arbitrary vertex weights. Proof: We need to show that set C is a vertex cover for G and w(C) < 2 ·OPT. Assume, for contradiction, that Cis not a vertex cover for G.

7 first finds a low cost Euler tour spanning the vertices of G, and then short-cuts this tour to find a traveling salesman tour. Is there a cheaper Euler tour than that found by doubling an MST? Recall that a graph has an Euler tour iff all its vertices have even degrees. Thus, we only need to be concerned about the vertices of odd degree in the MST. Let V' denote this set of vertices. IV'I must be even since the sum of degrees of all vertices in the MST is even. Now, if we add to the MST a minimum cost perfect matching on V', every vertex will have an even degree, and we get an Eulerian graph.

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