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标题：A Complete Characterization of the Approximability of Maximization Problems Derived from Boolean Constraint Satisfaction
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作者：Sanjeev Khanna ; Madhu Sudan ; David P. Williamson 等
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：1996
卷号：1996
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：In this paper we study the approximability of boolean constraint satisfaction problems. A problem in this class consists of some collection of ``constraints'' (i.e., functions f:01k01 ); an instance of a problem is a set of constraints applied to specified subsets of n boolean variables. Schaefer earlier studied the question of whether one could find in polynomial time a setting of the variables satisfying all constraints; he showed that every such problem is either in P or is NP-complete. We consider optimization variants of these problems in which one either tries to maximize the number of satisfied constraints (as in MAX 3SAT or MAX CUT) or tries to find an assignment satisfying all constraints which maximizes the number of variables set to 1 (as in MAX CUT or MAX CLIQUE). We completely classify the approximability of all such problems. In the first case, we show that any such optimization problem is either in P or is MAX SNP-hard. In the second case, we show that such problems fall precisely into one of five classes: solvable in polynomial-time, approximable to within constant factors in polynomial time (but no better), approximable to within polynomial factors in polynomial time (but no better), not approximable to within any factor but decidable in polynomial time, and not decidable in polynomial time (unless P = NP). This result proves formally for this class of problems two results which to this point have only been empirical observations; namely, that NP-hard problems in MAX SNP always turn out to be MAX SNP-hard, and that there seem to be no natural maximization problems approximable to within polylogarithmic factors but no better.