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.
- Austrin, P. and Håstad, J. 2011. Randomly supported independence and resistance. SIAM J. Comput. 40, 1, 1--27. Google Scholar
Digital Library
- Austrin, P. and Mossel, E. 2009. Approximation resistant predicates from pairwise independence. Comput. Complex. 18, 2, 249--271. Google Scholar
Digital Library
- Austrin, P., Benabbas, S., and Magen, A. 2010. On quadratic threshold csps. In Proceedings of the 9th Latin American Theoretical Informatics Symposium. A. López-Ortiz Ed., Springer, 332--343. Google Scholar
Digital Library
- Chow, C. K. 1961. On the characterization of threshold functions. In Proceedings of the 2nd Annual Symposium on Switching Circuit Theory and Logical Design. IEEE Computer Society, 34--38. Google Scholar
Digital Library
- Dertouzos, M. 1965. Threshold Logic: A Synthesis Approach. MIT Press, Cambridge, MA.Google Scholar
- Diakonikolas, I. and Servedio, R. A. 2009. Improved approximation of linear threshold functions. In Proceedings of the IEEE Conference on Computational Complexity. IEEE Computer Society, 161--172. Google Scholar
Digital Library
- Goemans, M. and Williamson, D. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115--1145. Google Scholar
Digital Library
- Hast, G. 2005. Beating a random assignment. In Proceedings of APPROX-RANDOM Conference. C. Chekuri, K. Jansen, J. D. P. Rolim, and L. Trevisan Eds., Springer, 134--145. Google Scholar
Digital Library
- Håstad, J. 2001. Some optimal inapproximability results. J. ACM 48, 798--859. Google Scholar
Digital Library
- Kaplan, K. R. and Winder, R. O. 1965. Chebyshev approximation and threshold functions. IEEE Trans. Electron. Comput. EC-14, 2, 250--252.Google Scholar
Cross Ref
- Kaszerman, P. 1963. A geometric test-synthesis procedure for a threshold device. Inf. Control 6, 4, 381--398.Google Scholar
Cross Ref
- Khot, S. 2002. On the power of unique 2-prover 1-round games. In Proceedings of the 34th ACM Symposium on Theory of Computating. J. H. Reif Ed., ACM, 767--775. Google Scholar
Digital Library
- Khot, S., Kindler, G., Mossel, E., and O’Donnell, R. 2007. Optimal inapproximability results for Max-Cut and other 2-variable CSPs? SIAM J. Comput. 37, 1, 319--357. Google Scholar
Digital Library
- O’Donnell, R. and Servedio, R. A. 2008. The Chow parameters problem. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. C. Dwork Ed., ACM, 517--526. Google Scholar
Digital Library
- O’Donnell, R. and Wu, Y. 2008. An optimal SDP algorithm for Max-Cut and equally optimal long code tests. In Proceedings of the 40th ACM Symposium on Theory of Computing. C. Dwork Ed., ACM, 335--344. Google Scholar
Digital Library
- Raghavendra, P. 2008. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. C. Dwork Ed., ACM, 245--254. Google Scholar
Digital Library
- Schaefer, T. 1978. The complexity of satisfiability problems. In Proceedings of the 10th ACM Symposium on Theory of Computing. R. J. Lipton, W. A. Burkhard, W. J. Savitch, E. P. Friedman, and A. V. Aho Eds., ACM, 216--226. Google Scholar
Digital Library
- Shiganov, I. S. 1986. Refinement of the upper bound of a constant in the remainder term of the central limit theorem. J. Soviet Math. 35, 109--115.Google Scholar
Cross Ref
- Winder, R. O. 1963. Threshold logic in artificial intelligence. IEEE Artif. Intell. S-142, 107--128.Google Scholar
- Winder, R. O. 1969. Threshold gate approximations based on Chow parameters. IEEE Trans. Comput. 18, 4, 372--375. Google Scholar
Digital Library
- Zwick, U. 1998. Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms. H. J. Karloff Ed., ACM/SIAM, 201--210. Google Scholar
Digital Library
Index Terms
Approximating Linear Threshold Predicates
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Approximating linear threshold predicates
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