, Volume 18, Issue 3, pp 307–343 | Cite as

Multiset variable representations and constraint propagation

  • Y. C. Law
  • J. H. M. Lee
  • T. Walsh
  • M. H. C. Woo


Multisets generalize sets by allowing elements to have repetitions. In this paper, we study from a formal perspective representations of multiset variables, and the consistency and propagation of constraints involving multiset variables. These help us model problems more naturally and can, for example, prevent introducing unnecessary symmetries into a model. We identify a number of different representations for multiset variables, compare them in terms of effectiveness and efficiency, and propose inference rules to enforce bounds consistency for the representations. In addition, we propose to exploit the variety of a multiset—the number of distinct elements in it—to improve modeling expressiveness and further enhance constraint propagation. We derive a number of inference rules involving the varieties of multiset variables. The rules interact varieties with the traditional components of multiset variables (such as cardinalities) to obtain stronger propagation. We also demonstrate how to apply the rules to perform variety reasoning on some common multiset constraints. Experimental results show that performing variety reasoning on top of cardinality reasoning can effectively reduce more search space and achieve better runtime in solving some multiset CSPs.


Constraint satisfaction Multiset variables 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azevedo, F. (2007). Cardinal: a finite sets constraint solver. Constraints, 12(1), 93–129.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Azevedo, F., & Barahona, P. (2000). Modelling digital circuits problems with set constraints. In Proceedings of the 1st international conference on computational logic (pp. 414–428).Google Scholar
  3. 3.
    Bennett, F.E., & Mendelsohn, E. (1980). Extended (2, 4)-designs. Journal of Combinatorial Theory, Series A, 29(1), 74–86.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Debruyne, R., & Bessière, C. (1997). Some practicable filtering techniques for the constraint satisfaction problem. In Proceedings of the 15th international joint conference on artificial intelligence (pp. 412–417).Google Scholar
  5. 5.
    Gecode Team (2006). Gecode: Generic constraint development environment. Available from
  6. 6.
    Gent, I.P., & Walsh, T. (1999). CSPLib: a benchmark library for constraints. Technical report, Technical report APES-09-1999. A shorter version appears in Proceedings of the 5th international conference on principles and practices of constraint programming.Google Scholar
  7. 7.
    Gervet, C. (1994). Conjunto: constraint logic programming with finite set domains. In Proceedings of the 1994 symposium on logic programming (pp. 339–358).Google Scholar
  8. 8.
    Gervet, C. (1997). Interval propagation to reason about sets: definition and implementation of a practical language. Constraints, 1(3), 191–244.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gervet, C., & Van Hentenryck, P. (2006). Length-lex ordering for set csps. In Proceedings of the 21st national conference on artificial intelligence (pp. 48–53).Google Scholar
  10. 10.
    Hawkins, P., Lagoon, V., Stuckey, P.J. (2005). Solving set constraint satisfaction problems using ROBDDs. Journal of Artificial Intelligence Research, 24, 109–156.CrossRefzbMATHGoogle Scholar
  11. 11.
    ILOG (2003). ILOG solver 6.0 reference manual.Google Scholar
  12. 12.
    Johnson, D.M., & Mendelsohn, N.S. (1972). Extended triple systems. Aequationes Mathematicae, 8(3), 291–298.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laburthe, F. (2000). Choco: implementing a CP kernel. In CP00 post conference workshop on techniques for implementing constraint programming systems (TRICS) (pp. 71–85).Google Scholar
  14. 14.
    Lagoon, V., & Stuckey, P.J. (2004). Set domain propagation using ROBDDs. In Proceedings of the 10th international conference on principles and practice of constraint programming (pp. 347–361).Google Scholar
  15. 15.
    Law, Y.C., Lee, J.H.M., Woo, M.H.C. (2009). Variety reasoning for multiset constraint propagation. In Proceedings of the 21st international joint conference on artificial intelligence (pp. 552–558).Google Scholar
  16. 16.
    Mackworth, A.K. (1977). Consistency in networks of relations. Artificial Intelligence, 8(1), 99–118.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mohr, R., & Masini, G. (1988). Good old discrete relaxation. In Proceedings of the 8th European conference on artificial intelligence (pp. 651–656).Google Scholar
  18. 18.
    Müller, T. (2001). Constraint propagation in Mozart. Doctoral dissertation, Universität des Saarlandes, Germany.Google Scholar
  19. 19.
    Müller, T., & Müller, M. (1997). Finite set constraints in Oz. In 13. Workshop logische programmierung (pp. 104–115).Google Scholar
  20. 20.
    Nadel, B.A. (1989). Constraint satisfaction algorithms. Computational Intelligence, 5, 188–224.CrossRefGoogle Scholar
  21. 21.
    Park, E.Y., & Blake, I. (2008). Construction of extended Steiner systems for information retrieval. Revista Matemática Complutense, 21(1), 179–190.MathSciNetzbMATHGoogle Scholar
  22. 22.
    Walsh, T. (2003). Consistency and propagation with multiset constraints: a formal viewpoint. In Proceedings of the 9th international conference on principles and practice of constraint programming (pp. 724–738).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Y. C. Law
    • 1
  • J. H. M. Lee
    • 1
  • T. Walsh
    • 2
  • M. H. C. Woo
    • 1
  1. 1.Department of Computer Science and EngineeringThe Chinese University of Hong KongShatin, N.T.Hong Kong
  2. 2.NICTA & UNSWSydneyAustralia

Personalised recommendations