Approximation Algorithms and Semidefinite Programming
Semidefinite programs constitute one of the largest classes of optimization problems that can be solved with reasonable efficiency - both in theory and practice. They play a key role in a variety of research areas, such as combinatorial optimization, appr
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Bernd G¨artner
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Jiˇr´ı Matouˇsek
Approximation Algorithms and Semidefinite Programming
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Bernd G¨artner ETH Zurich Institute of Theoretical Computer Science 8092 Zurich Switzerland [email protected]
Jiˇr´ı Matouˇsek Charles University Department of Applied Mathematics Malostransk´e n´am. 25 118 00 Prague 1 Czech Republic [email protected]
ISBN 978-3-642-22014-2 e-ISBN 978-3-642-22015-9 DOI 10.1007/978-3-642-22015-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011943166 Mathematics Subject Classification (2010): 68W25, 90C22 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
This text, based on a graduate course taught by the authors, introduces the reader to selected aspects of semidefinite programming and its use in approximation algorithms. It covers the basics as well as a significant amount of recent and more advanced material, sometimes on the edge of current research. Methods based on semidefinite programming have been the big thing in optimization since the 1990s, just as methods based on linear programming had been the big thing before that – at least this seems to be a reasonable picture from the point of view of a computer scientist. Semidefinite programs constitute one of the largest classes of optimization problems that can be solved reasonably efficiently – both in theory and in practice. They play an important role in a variety of research areas, such as combinatorial optimization, approximation algorithms, computational complexity, graph theory, geometry, real algebraic geometry, and quantum computing. We develop the basic theory of semidefinite programming; we present one of the known efficient algorithms in detail, and we describe the principles of some others. As for applications, we focus on approximation algorithms. There are many important computational problems, such as MaxCut,1 for which one cannot expect to obtain an exact solution efficiently, and in such cases one has to settle for approximate solutions. The main theoretical goal in this situation is to find efficient (polynomialtime) algorithms that always co
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