Is Constraint Satisfaction Over Two Variables Always Easy?
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By the breakthrough work of Håstad, several constraint satisfaction problems are now known to have the following approximation resistance property: satisfying more clauses than what picking a random assignment would achieve is NP-hard. This is the case for example for Max E3-Sat, Max E3-Lin and Max E4-Set Splitting. A notable exception to this extreme hardness is constraint satisfaction over two variables (2-CSP); as a corollary of the celebrated Goemans-Williamson algorithm, we know that every Boolean 2-CSP has a non-trivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables. An intriguing question then is whether this is also the case for 2-CSPs over larger, non-Boolean domains. This question is still open, and is equivalent to whether the generalization of Max 2-SAT to domains of size d, can be approximated to a factor better than (1 - x1/d 2).
In an attempt to make progress towards this question, in this paper we prove, firstly, that a slight restriction of this problem, namely a generalization of linear inequations with two variables per constraint, is not approximation resistant, and, secondly, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3. In the Boolean case, Not-All-Equal Sat with three variables per constraint is equivalent to Max 2-SAT and thus has a non-trivial approximation algorithm; for larger domain sizes, Max 2-SAT can be reduced to Not-All-Equal Sat with three variables per constraint. Our approximation algorithm implies that a wide class of 2-CSPs called regular 2-CSPs can all be approximated beyond their random assignment threshold.
KeywordsApproximation Algorithm Constraint Satisfaction Constraint Satisfaction Problem Performance Ratio Hardness Result
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- 1.Gunnar Andersson. Some New Randomized Approximation Algorithms. Doctoral dissertation, Department of Numerical Analysis and Computer Science, Royal Institute of Technology, May 2000.Google Scholar
- 2.Gunnar Andersson, Lars Engebretsen, and Johan Håstad. A new way of using semidefinite programming with applications to linear equations mod p. Journal of Algorithms, 39(2):162–204, May 2001.Google Scholar
- 4.Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115–1145, November 1995.Google Scholar
- 5.Michel X. Goemans and David P. Williamson. Approximation algorithms for Max-3-Cut and other problems via complex semidefinite programming. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 443–452. Hersonissos, Crete, Grece, 6–8 July 2001.Google Scholar
- 6.Johan Håstad. Some optimal inapproximability results. Journal of the ACM, 48(4):798–859, July 2001.Google Scholar
- 7.Viggo Kann, Sanjeev Khanna, Jens Lagergren, and Alessandro Panconesi. On the hardness of approximating Max k-Cut and its dual. Chicago Journal of Theoretical Computer Science, 1997(2), June 1997.Google Scholar
- 8.Subhash Khot. Hardness results for coloring 3-colorable 3-uniform hypergraphs. To appear in Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science. Vancouver, Canada, 16-19 November 2002.Google Scholar
- 9.Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 767–775. Montréal, Québec, Canada, 19–21 May 2002.Google Scholar
- 11.Uri Zwick. Approximation algorithms for constraint satisfaction programs involving at most three variables per constraint. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 201–210. San Francisco, California, 25-27 January 1998.Google Scholar
- 12.Uri Zwick. Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 679–687. Atlanta, Georgia, 1-4 May 1999.Google Scholar