Approximating Linear Threshold Predicates
Paper i proceeding, 2010

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(w1 x1+⋯+wn x n ) for some positive integer weights w 1, ..., w 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 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(x 1+⋯+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.

linear threshold predicates

constraint satisfaction problems



M. Cheraghchi

Ecole Polytechnique Federale De Lausanne

J. Håstad

Kungliga Tekniska Högskolan (KTH)

Marcus Isaksson

Chalmers, Matematiska vetenskaper, matematisk statistik

Göteborgs universitet

O. Svensson

Kungliga Tekniska Högskolan (KTH)

Lecture Notes in Computer Science

0302-9743 (ISSN)

Vol. 6302 110-123