Abstract
Max-Sur-CSP is the following optimisation problem: given a set of constraints, find a surjective mapping of the variables to domain values that satisfies as many of the constraints as possible. Many natural problems, e.g. Minimum k-Cut (which has many different applications in a variety of fields) and Minimum Distance (which is an important problem in coding theory), can be expressed as Max-Sur-CSPs. We study Max-Sur-CSP on the two-element domain and determine the computational complexity for all constraint languages (families of allowed constraints). Our results show that the problem is solvable in polynomial time if the constraint language belongs to one of three classes, and NP-hard otherwise. An important part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised minimum cut problem. This algorithm may be of independent interest.
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References
Bach, W., Zhou, H.: Approximation for maximum surjective constraint satisfaction problems. arXiv:1110.2953v1 [cs.CC] (2011)
Bodirsky, M., Kára, J., Martin, B.: The complexity of surjective homomorphism problems – a survey. Discrete Applied Mathematics 160(12), 1680–1690 (2012)
Bulatov, A.A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53, 66–120 (2006)
Bulatov, A.A., Dalmau, V., Grohe, M., Marx, D.: Enumerating homomorphisms. J. Comput. Syst. Sci. 78(2), 638–650 (2012)
Cohen, D.A.: Tractable decision for a constraint language implies tractable search. Constraints 9, 219–229 (2004)
Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: Supermodular functions and the complexity of max CSP. Discrete Applied Mathematics 149(1-3), 53–72 (2005)
Cohen, D.A., Cooper, M.C., Jeavons, P.G., Krokhin, A.A.: The complexity of soft constraint satisfaction. Artif. Intell. 170, 983–1016 (2006)
Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. J. Comput. Syst. Sci. 51, 511–522 (1995)
Creignou, N., Hébrard, J.J.: On generating all solutions of generalized satisfiability problems. Informatique Théorique et Applications 31(6), 499–511 (1997)
Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Information and Computation 125(1), 1–12 (1996)
Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of boolean constraint satisfaction problems. Society for Industrial and Applied Mathematics, Philadelphia (2001)
Dechter, R., Itai, A.: Finding all solutions if you can find one. In: AAAI 1992 Workshop on Tractable Reasoning, pp. 35–39 (1992)
Golovach, P.A., Paulusma, D., Song, J.: Computing Vertex-Surjective Homomorphisms to Partially Reflexive Trees. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 261–274. Springer, Heidelberg (2011)
Horn, A.: On sentences which are true of direct unions of algebras. The Journal of Symbolic Logic 16(1), 14–21 (1951)
Jonsson, P., Klasson, M., Krokhin, A.: The approximability of three-valued max CSP. SIAM J. Comput. 35, 1329–1349 (2006)
Jonsson, P., Kuivinen, F., Thapper, J.: Min CSP on Four Elements: Moving beyond Submodularity. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 438–453. Springer, Heidelberg (2011)
Karger, D.R.: Global min-cuts in RNC, and other ramifications of a simple min-out algorithm. In: Proceedings of the Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 1993, pp. 21–30. Society for Industrial and Applied Mathematics, Philadelphia (1993)
Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30, 1863–1920 (2001)
Kolmogorov, V., Živný, S.: The complexity of conservative valued CSPs. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 750–759. SIAM (2012)
Martin, B., Paulusma, D.: The Computational Complexity of Disconnected Cut and 2K2-Partition. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 561–575. Springer, Heidelberg (2011)
Nagamochi, H., Nishimura, K., Ibaraki, T.: Computing all small cuts in an undirected network. SIAM J. Discret. Math. 10, 469–481 (1997)
Golovach, P.A., Lidický, B., Martin, B., Paulusma, D.: Finding vertex-surjective graph homomorphisms. arXiv:1204.2124v1 [cs.DM] (2012)
Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC 2008, pp. 245–254. ACM, New York (2008)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 1978, pp. 216–226. ACM, New York (1978)
Valiant, L.G.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)
Vardy, A.: Algorithmic complexity in coding theory and the minimum distance problem. In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, STOC 1997, pp. 92–109. ACM, New York (1997)
Vazirani, V., Yannakakis, M.: Suboptimal Cuts: Their Enumeration, Weight and Number. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 366–377. Springer, Heidelberg (1992)
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Uppman, H. (2012). Max-Sur-CSP on Two Elements. In: Milano, M. (eds) Principles and Practice of Constraint Programming. CP 2012. Lecture Notes in Computer Science, vol 7514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33558-7_6
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DOI: https://doi.org/10.1007/978-3-642-33558-7_6
Publisher Name: Springer, Berlin, Heidelberg
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