Abstract
Max CSP( P ) is the problem of maximizing the weight of satisfied constraints, where each constraint acts over a k-tuple of literals and is evaluated using the predicate P. The approximation ratio of a random assignment is equal to the fraction of satisfying inputs to P. If it is NP-hard to achieve a better approximation ratio for Max CSP( P ), then we say that P is approximation resistant. Our goal is to characterize which predicates that have this property.
A general approximation algorithm for Max CSP( P ) is introduced. For a multitude of different P, it is shown that the algorithm beats the random assignment algorithm, thus implying that P is not approximation resistant. In particular, over 2/3 of the predicates on four binary inputs are proved not to be approximation resistant, as well as all predicates on 2s binary inputs, that have at most 2s+1 accepting inputs.
We also prove a large number of predicates to be approximation resistant. In particular, all predicates of arity 2s+s 2 with less than \(2^{s^2}\) non-accepting inputs are proved to be approximation resistant, as well as almost 1/5 of the predicates on four binary inputs.
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Hast, G. (2005). Beating a Random Assignment. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_12
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DOI: https://doi.org/10.1007/11538462_12
Publisher Name: Springer, Berlin, Heidelberg
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