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Some optimal inapproximability results Johan H˚ astad Royal Institute of Technology Sweden email:[email protected] September 4, 1998 Abstract We prove optimal, up to an arbitrary ǫ > 0, inapproximability results for Max-E k -Sat 1 for k 3, maximizing the number of satisfied linear equa- tions in an overdetermined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-Di-Cut, and Vertex cover. For Max-E2-Sat the obtained lower bound is essentially 22 / 21 1 . 047 while the strongest upper bound is around 1 . 074. 1 Introduction We know that many natural optimization problems are NP-hard. This means that they are probably hard to solve exactly in the worst case. In practice, however, it is many times sufficient to get reasonable good solutions for all (or even most) instances. In this paper we study the existence of polynomial time approximation algorithms for some of the basic NP-complete problems. We say that an algorithm is a C -approximation algorithm if it for each instance produces an answer that is at most off by a factor C from the optimal answer. The fundamental question is for a given NP-complete problem, for what value of C can we hope for a polynomial time C -approximation algorithm. Posed in this generality this is a large research area with many positive and negative results. In this paper we concentrate on negative results, i.e. results of the form that for some C > 1 a certain problem cannot be approximated within C in polynomial time. These results are invariably based on plausible complexity theoretic assumptions, the weakest possible being NP negationslash =P since if NP=P, all considered problems can be solved exactly in polynomial time. The most basic NP-complete problem is satisfiability of CNF-formulas and probably the most used variant of this is 3-SAT where each clause contains at most 3 variables. For simplicity, let us assume that each clause contains exactly 1 Max-E k -Sat is the variant of CNF-Sat where each clause is of length exactly k . 1
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3 variables. The optimization variant of this problem is to satisfy as many clauses as possible. It is not hard to see that a random assignment satisfies each clause with probability 7/8 and hence if there are m clauses it is not hard (even deterministically) to find an assignment that satisfies 7 m/ 8 clauses. Since we can never satisfy more than all the clauses this gives a 8 / 7-approximation algorithm. This was one of the first approximation algorithms considered [17] and one of the main results of this paper is that this is optimal to within an arbitrary additive constant ǫ .
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