Towards a Characterization of Approximation Resistance for Symmetric CSPs

Authors Venkatesan Guruswami, Euiwoong Lee

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Venkatesan Guruswami
Euiwoong Lee

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Venkatesan Guruswami and Euiwoong Lee. Towards a Characterization of Approximation Resistance for Symmetric CSPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 305-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


A Boolean constraint satisfaction problem (CSP) is called approximation resistant if independently setting variables to 1 with some probability achieves the best possible approximation ratio for the fraction of constraints satisfied. We study approximation resistance of a natural subclass of CSPs that we call Symmetric Constraint Satisfaction Problems (SCSPs), where satisfaction of each constraint only depends on the number of true literals in its scope. Thus a SCSP of arity k can be described by a subset of allowed number of true literals. For SCSPs without negation, we conjecture that a simple sufficient condition to be approximation resistant by Austrin and Hastad is indeed necessary. We show that this condition has a compact analytic representation in the case of symmetric CSPs (depending only on the gap between the largest and smallest numbers in S), and provide the rationale behind our conjecture. We prove two interesting special cases of the conjecture, (i) when S is an interval and (ii) when S is even. For SCSPs with negation, we prove that the analogous sufficient condition by Austrin and Mossel is necessary for the same two cases, though we do not pose an analogous conjecture in general.
  • Constraint Satisfaction Problems
  • Approximation resistance


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  1. Per Austrin, Siavosh Benabbas, and Avner Magen. On quadratic threshold CSPs. Discrete Mathematics & Theoretical Computer Science, 14(2):205-228, 2012. Google Scholar
  2. Per Austrin and Johan Håstad. On the usefulness of predicates. ACM Transactions on Computation Theory, 5(1):1:1-1:24, May 2013. Google Scholar
  3. Per Austrin and Subhash Khot. A characterization of approximation resistance for even k-partite CSPs. In Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, ITCS'13, pages 187-196, 2013. Google Scholar
  4. Per Austrin and Elchanan Mossel. Approximation resistant predicates from pairwise independence. Computational Complexity, 18(2):249-271, 2009. Google Scholar
  5. Boaz Barak, Siu On Chan, and Pravesh Kothari. Sum of squares lower bounds from pairwise independence. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing, STOC'15, 2015. To appear. Google Scholar
  6. Siavosh Benabbas, Konstantinos Georgiou, Avner Magen, and Madhur Tulsiani. SDP gaps from pairwise independence. Theory of Computing, 8(1):269-289, 2012. Google Scholar
  7. Siu On Chan. Approximation resistance from pairwise independent subgroups. In Proceedings of the 45th annual ACM Symposium on Theory of Computing, STOC'13, pages 447-456, 2013. Google Scholar
  8. Mahdi Cheraghchi, Johan Håstad, Marcus Isaksson, and Ola Svensson. Approximating linear threshold predicates. ACM Trans. Comput. Theory, 4(1):2:1-2:31, 2012. Google Scholar
  9. Wing-Sum Cheung. Generalizations of Hölder’s inequality. International Journal of Mathematics and Mathematical Sciences, 26(1):7-10, 2001. Google Scholar
  10. G Hast. Beating a random assignment. KTH, Stockholm. PhD thesis, Ph. D Thesis, 2005. Google Scholar
  11. Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798-859, July 2001. Google Scholar
  12. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th annual ACM Symposium on Theory of Computing, STOC'02, pages 767-775, 2002. Google Scholar
  13. Subhash Khot, Madhur Tulsiani, and Pratik Worah. A characterization of strong approximation resistance. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, STOC'14, pages 634-643, 2014. Google Scholar
  14. Thomas Schaefer. The complexity of satisfiability problems. In Proceedings of the 10th annual ACM Symposium on Theory of Computing, STOC'78, pages 216-226, 1978. Google Scholar
  15. Madhur Tulsiani. CSP gaps and reductions in the Lasserre hierarchy. In Proceedings of the 41st annual ACM Symposium on Theory of Computing, STOC'09, pages 303-312, 2009. Google Scholar
  16. Uri Zwick. 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, SODA'98, pages 201-210, 1998. Google Scholar
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