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

Keywords

Polynomial Time Variable Domain Domain Versus Global Constraint Cardinality Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beldiceanu, N., Contejean, E.: Introducing Global Constraints in CHIP. Journal of Mathematical and Computer Moddeling 20(12), 97–123 (1994)MATHCrossRefGoogle Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)Google Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Dechter, R., Meiri, I., Pearl, J.: Temporal constraint networks. Artificial Intelligence 49(1-3), 61–95 (1991)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Katriel, I., Thiel, S.: Complete Bound Consistency for the Global Cardinality Constraint. Constraints 10(3), 191–217 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Lawler, E.: Combinatorial Optimization: Networks and Matroids. Saunders College Publishing (1976)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Régin, J.-C.: Generalized Arc Consistency for Global Cardinality Constraint. In: 13th Conference on Artificial Intelligence, AAAI 1996, pp. 209–215 (1996)Google Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Shostak, R.: Deciding linear inequalities by computing loop residues. Journal of the ACM 28(4), 769–779 (1981)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    CHOCO Team. choco: an open source java constraint programming library. Research report 10-02-INFO, Ecole des Mines de Nantes (2010)Google Scholar
  15. 15.
    van Hoeve, W.-J., Pesant, G., Rousseau, L.-M., Sabharwal, A.: New filtering algorithms for combinations of among constraints. Constraints 14, 273–292 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Gilles Chabert
    • 1
  • Sophie Demassey
    • 1
  1. 1.TASC, Mines-Nantes, INRIA, LINA CNRSNantesFrance

Personalised recommendations