Approximating Linear Threshold Predicates

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 "majoritylike" 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.