Approximating Linear Threshold Predicates

Abstract

We study constraint satisfaction problems on the domain {−1, 1}, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form sgn(w1x1 + · · · + wnxn) for some positive integer weights w1,..., wn. Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not. The focus of this paper is to identify and study the approximation curve of a class of threshold predicates that allow for non-trivial approximation. Arguably the simplest such predicate is the majority predicate sgn(x1 + · · · + xn), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of “majority-like ” predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest

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