7/8-Approximation algorithm for MAX 3SAT?

Howard Karloff, Uri Zwick

Research output: Contribution to journalConference articlepeer-review

Abstract

We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instance - a collection of clauses each of length at most three - is satisfiable, then the expected weight of the assignment found is at least 7/8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAX CUT algorithm of Goemans and Williamson and the MAX 2SAT algorithm of Feige and Goemans. Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P≠NP, no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7/8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semidefinite relaxations of any constraint satisfaction problem of the form MAX CSP(F), where F is a finite family of Boolean functions. Our relaxations are the strongest possible within a natural class of semidefinite relaxations.

Original languageEnglish
Pages (from-to)406-415
Number of pages10
JournalAnnual Symposium on Foundations of Computer Science - Proceedings
StatePublished - 1997
Externally publishedYes
EventProceedings of the 1997 38th IEEE Annual Symposium on Foundations of Computer Science - Miami Beach, FL, USA
Duration: 20 Oct 199722 Oct 1997

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