# Advanced Algorithms Essay

College of Computing Georgia Institute of Technology

Copyright c 2001

Approximation Algorithms

Springer

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To my parents

Preface

Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872–1970) Most natural optimization problems, including those arising in important application areas, are NP-hard. Therefore, under the widely believed conjecture that P = NP, their exact solution is prohibitively time consuming. Charting the landscape of approximability of these problems, via polynomial time algorithms, therefore becomes a compelling subject of scientiﬁc inquiry in computer

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However, there is no theorem to this eﬀect yet.

VIII

Preface

taining eﬃcient approximate counting algorithms for this latter class of problems. Most of these algorithms are based on the Markov chain Monte Carlo (MCMC) method, a topic that deserves a book by itself and is therefore not treated here. In Chapter 28 we present combinatorial algorithms, not using the MCMC method, for two fundamental counting problems. The third topic is centered around recent breakthrough results, establishing hardness of approximation for many key problems, and giving new legitimacy to approximation algorithms as a deep theory. An overview of these results is presented in Chapter 29, assuming the main technical theorem, the PCP Theorem. The latter theorem, unfortunately, does not have a simple proof at present. The fourth topic consists of the numerous open