Approximating linear threshold predicates

Authors:
Mahdi Cheraghchi;Johan Håstad;Marcus Isaksson;Ola Svensson
Affiliations:
School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland;Royal Institute of Technology, Stockholm, Sweden;Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden;Royal Institute of Technology, Stockholm, Sweden
Venue:
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Year:
2010

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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 "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.