Abstract
An Among constraint holds if the number of variables that belong to a given value domain is between given bounds. This paper focuses on the case where the variable and value domains are intervals. We investigate the conjunction of Among constraints of this type. We prove that checking for satisfiability – and thus, enforcing bound consistency – can be done in polynomial time. The proof is based on a specific decomposition that can be used as such to filter inconsistent bounds from the variable domains. We show that this decomposition is incomparable with the natural conjunction of Among constraints, and that both decompositions do not ensure bound consistency. Still, experiments on randomly generated instances reveal the benefits of this new decomposition in practice. This paper also introduces a generalization of this problem to several dimensions and shows that satisfiability is \(\mathcal{R}\)-complete in the multi-dimensional case
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References
Beldiceanu, N., Contejean, E.: Introducing Global Constraints in CHIP. Journal of Mathematical and Computer Moddeling 20(12), 97–123 (1994)
Bessière, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: Among, Common and Disjoint Constraints. In: Hnich, B., Carlsson, M., Fages, F., Rossi, F. (eds.) CSCLP 2005. LNCS (LNAI), vol. 3978, pp. 29–43. Springer, Heidelberg (2006)
Bessière, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: The Range and Roots Constraints: Specifying Counting and Occurrence Problems. In: IJCAI, pp. 60–65 (2005)
Brand, S., Narodytska, N., Quimper, C.-G., Stuckey, P.J., Walsh, T.: Encodings of the Sequence Constraint. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 210–224. Springer, Heidelberg (2007)
Chabert, G., Jaulin, L., Lorca, X.: A Constraint on the Number of Distinct Vectors with Application to Localization. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 196–210. Springer, Heidelberg (2009)
Cotton, S., Maler, O.: Fast and Flexible Difference Constraint Propagation for DPLL(T). In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 170–183. Springer, Heidelberg (2006)
Dechter, R., Meiri, I., Pearl, J.: Temporal constraint networks. Artificial Intelligence 49(1-3), 61–95 (1991)
Katriel, I., Thiel, S.: Complete Bound Consistency for the Global Cardinality Constraint. Constraints 10(3), 191–217 (2005)
Lawler, E.: Combinatorial Optimization: Networks and Matroids. Saunders College Publishing (1976)
Maher, M.J., Narodytska, N., Quimper, C.-G., Walsh, T.: Flow-Based Propagators for the SEQUENCE and Related Global Constraints. In: Stuckey, P.J. (ed.) CP 2008. LNCS, vol. 5202, pp. 159–174. Springer, Heidelberg (2008)
Régin, J.-C.: Generalized Arc Consistency for Global Cardinality Constraint. In: 13th Conference on Artificial Intelligence, AAAI 1996, pp. 209–215 (1996)
Régin, J.-C.: Combination of Among and Cardinality Constraints. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 288–303. Springer, Heidelberg (2005)
Shostak, R.: Deciding linear inequalities by computing loop residues. Journal of the ACM 28(4), 769–779 (1981)
CHOCO Team. choco: an open source java constraint programming library. Research report 10-02-INFO, Ecole des Mines de Nantes (2010)
van Hoeve, W.-J., Pesant, G., Rousseau, L.-M., Sabharwal, A.: New filtering algorithms for combinations of among constraints. Constraints 14, 273–292 (2009)
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Chabert, G., Demassey, S. (2012). The Conjunction of Interval Among Constraints. In: Beldiceanu, N., Jussien, N., Pinson, É. (eds) Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems. CPAIOR 2012. Lecture Notes in Computer Science, vol 7298. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29828-8_8
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DOI: https://doi.org/10.1007/978-3-642-29828-8_8
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