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Unformatted text preview: Some optimal inapproximability results Johan H astad Royal Institute of Technology Sweden email:johanh@nada.kth.se September 4, 1998 Abstract We prove optimal, up to an arbitrary > 0, inapproximability results for MaxE kSat 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 MaxE2Sat, MaxCut, MaxDiCut, and Vertex cover. For MaxE2Sat 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 NPhard. 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 NPcomplete problems. We say that an algorithm is a Capproximation 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 NPcomplete problem, for what value of C can we hope for a polynomial time Capproximation 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 NPcomplete problem is satisfiability of CNFformulas and probably the most used variant of this is 3SAT where each clause contains at most 3 variables. For simplicity, let us assume that each clause contains exactly 1 MaxE kSat is the variant of CNFSat where each clause is of length exactly k . 1 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 / 7approximation 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 ....
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This note was uploaded on 11/21/2011 for the course CSCI 2921 taught by Professor Zgai during the Spring '11 term at Minnesota.
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