By Vijay V. Vazirani

This e-book covers the dominant theoretical methods to the approximate answer of tough combinatorial optimization and enumeration difficulties. It comprises dependent combinatorial thought, worthwhile and fascinating algorithms, and deep effects concerning the intrinsic complexity of combinatorial difficulties. Its readability of exposition and ideal collection of routines will make it obtainable and attractive to all people with a flavor for arithmetic and algorithms.

Richard Karp,University Professor, college of California at Berkeley

Following the improvement of uncomplicated combinatorial optimization thoughts within the Nineteen Sixties and Nineteen Seventies, a major open query used to be to improve a thought of approximation algorithms. within the Nineteen Nineties, parallel advancements in options for designing approximation algorithms in addition to equipment for proving hardness of approximation effects have ended in a gorgeous thought. the necessity to remedy really huge cases of computationally tough difficulties, equivalent to these coming up from the web or the human genome venture, has additionally elevated curiosity during this concept. the sphere is at present very energetic, with the toolbox of approximation set of rules layout strategies getting continuously richer.

It is a excitement to suggest Vijay Vazirani's well-written and finished publication in this vital and well timed subject. i'm certain the reader will locate it Most worthy either as an creation to approximability in addition to a connection with the numerous elements of approximation algorithms.

László Lovász, Senior Researcher, Microsoft Research

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**Extra info for Approximation Algorithms**

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Use the factor f set cover algorithm. 15 (Hochbaum [132]) Consider the following problem. 18 (Maximum coverage) Given a universal set U of n elements, with nonnegative weights specified, a collection of subsets of U, sl. 'sl, and an integer k, pick k sets so as to maximize the weight of elements covered. 16 Using set cover, obtain approximation algorithms for the following variants of the shortest superstring problem (here sR is the reverse of string s): 1. Find the shortest string that contains, for each string Si E S, both Si and sf as substrings.

The weight of a minimum u-v cut in G is w' (u, v). Since Ai is a u-v cut in G, w(Ai) ~ w'(u,v). Thus each cut among A 1 , A 2 , ... , Ak_ 1 is at least as heavy as the cut defined in G by the corresponding edge of B'. This, together with the fact that C is the union of the lightest k- 1 cuts defined by T, gives: w(C) :':::: e~' w'(e) :':::: t; w(Ai) :':::: k-1 ( 1- 1) t; k k w(Ai) ( 1) = 2 1- k w(A). 9 The tight example given above for multiway cuts on 2k vertices also serves as a tight example for the k-cut algorithm (of course, there is no need to mark vertices as terminals).

Ei}· A dominating set in an undirected graph H = (U,F) is a subset S s;;: U such that every vertex in U-S is adjacent to a vertex in S. Let dom(H) denote the size of a minimum cardinality dominating set in H. Computing dom(H) is NP-hard. , Gi contains k stars spanning all vertices, where a star is the graph K 1,p, with p ~ 1. If i* is the smallest such index, then cost(ei*) is the cost of an optimal k-center. We will denoted this by OPT. We will work with the family of graphs G1, ... , Gm. Define the square of graph H to be the graph containing an edge (u, v) whenever H has a path of length at most two between u and v, u /:- v.