Abstract
The maximum constraint satisfaction problem (Max CSP) is the following computational problem: an instance is a finite collection of constraints on a set of variables, and the goal is to assign values to the variables that maximises the number of satisfied constraints. Max CSP captures many well-known problems (such as Max k -SAT and Max Cut) and so is NP-hard in general. It is natural to study how restrictions on the allowed constraint types (or constraint language) affect the complexity and approximability of Max CSP. All constraint languages, for which the CSP problem (i.e., the problem of deciding whether all constraints in an instance can be simultaneously satisfied) is currently known to be NP-hard, have a certain algebraic property, and it has been conjectured that CSP problems are tractable for all other constraint languages. We prove that any constraint language with this algebraic property makes Max CSP hard at gap location 1, thus ruling out the existence of a polynomial-time approximation scheme for such problems. We then apply this result to Max CSP restricted to a single constraint type. We show that, unless P = NP, such problems either are trivial or else do not admit polynomial-time approximation schemes. All our hardness results hold even if the number of occurrences of each variable is bounded by a constant.
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References
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation: Combinatorial Optimization Problems and their Approximability Properties. Springer, Heidelberg (1999)
Bulatov, A.: Tractable conservative constraint satisfaction problems. In: LICS 2003. Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, pp. 321–330. IEEE Computer Society Press, Los Alamitos (2003)
Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53(1), 66–120 (2006)
Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34(3), 720–742 (2005)
Burris, S., Sankappanavar, H.: A Course in Universal Algebra. Springer, Heidelberg (1981)
Cohen, D., Cooper, M., Jeavons, P., Krokhin, A.: Supermodular functions and the complexity of Max CSP. Discrete Appl. Math. 149(1-3), 53–72 (2005)
Dalmau, V., Jeavons, P.: Learnability of quantified formulas. Theor. Comput. Sci. 306, 485–511 (2003)
Deineko, V., Jonsson, P., Klasson, M., Krokhin, A.: Supermodularity on chains and complexity of maximum constraint satisfaction. In: EuroComb 2005. European Conference on Combinatorics, Graph Theory and Applications, volume AE of DMTCS Proceedings, pp. 51–56, 2005. Discrete Mathematics and Theoretical Computer Science, Full version available as The approximability of Max CSP with fixed-value constraints, arXiv.org:cs.CC/0602075 (2005)
Feder, T., Hell, P., Huang, J.: List homomorphisms of graphs with bounded degrees. Discrete Math. 307, 386–392 (2007)
Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)
Jonsson, P., Klasson, M., Krokhin, A.: The approximability of three-valued Max CSP. SIAM J. Comput. 35(6), 1329–1349 (2006)
Jonsson, P., Krokhin, A.: Maximum H-colourable subdigraphs and constraint optimization with arbitrary weights. J. Comput. System Sci. 73(5), 691–702 (2007)
Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2000)
Krokhin, A., Larose, B.: Maximum constraint satisfaction on diamonds. Technical Report CS-RR-408, University of Warwick, UK (2004)
Krokhin, A., Larose, B.: Maximum constraint satisfaction on diamonds. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 388–402. Springer, Heidelberg (2005)
MacGillivray, G.: On the complexity of colouring by vertex-transitive and arc-transistive digraphs. SIAM J. Discret. Math. 4(3), 397–408 (1991)
Petrank, E.: The hardness of approximation: Gap location. Computational Complexity 4, 133–157 (1994)
Pöschel, R., Kalužnin, L.: Funktionen- und Relationenalgebren. DVW, Berlin (1979)
Rossi, F., van Beek, P., Walsh, T. (eds.): Handbook of Constraint Programming. Elsevier, Amsterdam (2006)
Trevisan, L.: Inapproximability of combinatorial optimization problems, arXiv.org:cs.CC/0409043 (2004)
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Jonsson, P., Krokhin, A., Kuivinen, F. (2007). Ruling Out Polynomial-Time Approximation Schemes for Hard Constraint Satisfaction Problems. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds) Computer Science – Theory and Applications. CSR 2007. Lecture Notes in Computer Science, vol 4649. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74510-5_20
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DOI: https://doi.org/10.1007/978-3-540-74510-5_20
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