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Gadgets, approximation, and linear programming

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We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804-915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1-10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16/17+ ∊ and 12/13+ ∊ , respectively, is NP-hard for every ∊ > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [SIAM J. Comput., 27 (1998), pp. 804-915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801.

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