Abstract
We study a family of problems, called Maximum Solution, where the objective is to maximise a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints. This problem is closely related to Integer Linear Programming.When the domain is Boolean (i.e. restricted to {0,1}), the maximum solution problem is identical to the well-studied Max Ones problem, and the approximability is completely understood for all restrictions on the underlying constraints. We continue this line of research by considering domains containing more than two elements. We present two main results: a complete classification for the approximability of all maximal constraint languages, and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints.Our results are proved by using algebraic results from clone theory and the results indicates that this approach is very useful for classifying the approximability of certain optimisation problems.
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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.: A dichotomy theorem for constraints on a three-element set. In: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS 2002), pp. 649–658 (2002)
Bulatov, A.: A graph of a relational structure and constraint satisfaction problems. In: Proceedings of the 19th IEEE Symposium on Logic in Computer Science (LICS 2004), pp. 448–457 (2004)
Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. Journal of the 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)
Bulatov, A., Krokhin, A., Jeavons, P.: The complexity of maximal constraint languages. In: Proceedings of the 33rd ACM Symposium on Theory of Computing (STOC 2001), pp. 667–674 (2001)
Cohen, D.A., Cooper, M., Jeavons, P.G., Krokhin, A.A.: Soft constraints: Complexity and multimorphisms. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 244–258. Springer, Heidelberg (2003)
Cohen, D.A.: Tractable decision for a constraint language implies tractable search. Constraints 9(3), 219–229 (2004)
Creignou, N., Khanna, S., Sudan, M.: Complexity classifications of Boolean constraint satisfaction problems. SIAM, Philadelphia (2001)
Dalmau, V.: A new tractable class of constraint satisfaction problems. Annals of Mathematics and Artificial Intelligence 44(1–2), 61–85 (2005)
Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Hochbaum, D.S., Naor, J.: Simple and fast algorithms for linear and integer programs with two variables per inequality. SIAM J. Comput. 23(6), 1179–1192 (1994)
Hoffmann, C.M.: Group-Theoretic Algorithms and Graph Isomorphism. LNCS, vol. 136. Springer, Heidelberg (1982)
Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the ACM 44, 527–548 (1997)
Jeavons, P., Cooper, M.: Tractable constraints on ordered domains. Artificial Intelligence 79, 327–339 (1996)
Jonsson, P.: Near-optimal nonapproximability results for some NPO PB-complete problems. Information Processing Letters 68(5), 249–253 (1998)
Jonsson, P., Klasson, M., Krokhin, A.: The approximability of three-valued Max CSP. SIAM J. Comput. 35(6), 1329–1349 (2006)
Jonsson, P., Kuivinen, F., Nordh, G.: Approximability of integer programming with generalised constraints. CoRR, cs.CC/0602047 (2006)
Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2001)
Kuivinen, F.: Tight approximability results for the maximum solution equation problem over z p . In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 628–639. Springer, Heidelberg (2005)
Pöschel, R., Kaluznin, L.: Funktionen- und Relationenalgebren. DVW, Berlin (1979)
Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons, Inc., New York (1986)
Szczepara, B.: Minimal clones generated by groupoids. Ph.D thesis, Université de Móntreal (1996)
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Jonsson, P., Kuivinen, F., Nordh, G. (2006). Approximability of Integer Programming with Generalised Constraints. In: Benhamou, F. (eds) Principles and Practice of Constraint Programming - CP 2006. CP 2006. Lecture Notes in Computer Science, vol 4204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11889205_20
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DOI: https://doi.org/10.1007/11889205_20
Publisher Name: Springer, Berlin, Heidelberg
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