GoeWil95 - Improved Maximum Approximation Algorithms for...

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Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming MIC13EL X. GOEMANS Massachusetts Institute of Technology, Cambridge, Massachusetts AND DAVID P. WILLIAMSON IBM T. J. Watson Research Center, Yorktown Heights, New York Abstract. We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had perfc~r- mance guarantees of ~ for MAX CUT and ~ for MAX 2SAT. Slight extensions of our analysis lead to a .79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a .758-approximation algorithm for MAX SAT, where the best previously known approxim a- tion algorithms had performance guarantees of ~ and ~, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of :semidefinite programming in the design of approximation algorithms. Categories and Subject Descriptors: F2.2 [Analysis of Algorithms and Problem Complexity]: Nonumerical Algorithms and Problems—computations on discrete structures; G2.2 [Discrete Math- A preliminary version has appeared in Proceedings of the 26th AnnualACM Symposium on Theory of Computing (Montreal, Que., Canada). ACM, New York, 1994, pp. 422–431. The research of M. X. Goemans was supported in part by National Science Foundation (NSF) contract CCR 93-02476 and DARPA contract NOO014-92-J-1799. The research of D. P. Williamson was supported by an NSF Postdoctoral Fellowship. This research was conducted while the author was visiting MIT. Authors’ addresses: M. X. Goemans, Department of Mathematics, Room 2-382, Massachusetts Institute of Technology, Cambridge, MA 02139, e-mail: [email protected] edu; D. P. Williamson, IBM T, J. Watson Research Center, Room 33-219, P.O. Box 218, Yorktown Heights, NY 1059I8, e-mail: [email protected] corn. Permission to make digital/hard copy of part or all of this work for personal or classroom use is grantedl without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given tlhat copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. 01995 ACM 0004-5411/95/1100-1115 $03.50 Journalof theAssociationfor ComputinsMachinery,Vol. 42,No.6,November1995,pp.1115-1145.
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1116 M. X. GOEMANS AND D. P. WILLIAMSON ematics]: Graph Theo~—graph algorithms; G3 [Probability and Statistics] —probabik~ic algo- rithms (including Monte-Car-lo); 11.2 [Algebraic manipulation]: Algorithms—analysis of algorithm General Terms: Algorithms Additional
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GoeWil95 - Improved Maximum Approximation Algorithms for...

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