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Approximating Linear Threshold Predicates

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

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(w 1 x 1 + ⋯ + w n 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 + ⋯ + x 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 Chow-robustness that might be of independent interest.

This research is supported by the ERC Advanced investigator grant 226203. M. Cheraghchi is supported by the ERC Advanced investigator grant 228021.

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Author information

Authors and Affiliations

  1. School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland

    Mahdi Cheraghchi

  2. KTH - Royal Institute of Technology, Stockholm, Sweden

    Johan Håstad & Ola Svensson

  3. Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

    Marcus Isaksson

Authors
  1. Mahdi Cheraghchi
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  2. Johan Håstad
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  3. Marcus Isaksson
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  4. Ola Svensson
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Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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Cheraghchi, M., Håstad, J., Isaksson, M., Svensson, O. (2010). Approximating Linear Threshold Predicates. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_9

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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

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