Approximating Linear Threshold Predicates

  • Authors:

  • Mahdi Cheraghchi;Johan Håstad;Marcus Isaksson;Ola Svensson

  • Affiliations:

  • Carnegie Mellon University;KTH Royal Institute of Technology;Chalmers University of Technology and University of Gothenburg;École Polytechnique Fédéral de Lausanne

  • Venue:
  • ACM Transactions on Computation Theory (TOCT)
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

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 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(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.