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Abstract:

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: f0; 1g k! f0; 1g); an instance of a problem is a set of constraints applied to specied subsets of n boolean variables. Schaefer earlier studied the question of whether one could nd 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 satis ed constraints (as in MAX 3SAT or MAX CUT) or tries to nd 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 rst 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 ve classes, assuming P 6 = NP: solvable in polynomialtime, 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. 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

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