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(<i>w</i>₁<i>x</i>₁ + ⋯ + <i>w</i>_n<i>x</i>_n) for some positive integer weights <i>w</i>₁, ..., <i>w</i>_n. 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 article is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate sgn(<i>x</i>₁ + ⋯ + <i>x</i>_n), 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 <i>Chow-robustness</i> that might be of independent interest.