The ROOTS Constraint

  • Christian Bessiere
  • Emmanuel Hebrard
  • Brahim Hnich
  • Zeynep Kiziltan
  • Toby Walsh
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4204)


A wide range of counting and occurrence constraints can be specified with just two global primitives: the Range constraint, which computes the range of values used by a sequence of variables, and the Roots constraint, which computes the variables mapping onto a set of values. We focus here on the Roots constraint. We show that propagating the Roots constraint completely is intractable. We therefore propose a decomposition which can be used to propagate the constraint in linear time. Interestingly, for all uses of the Roots constraint we have met, this decomposition does not destroy the global nature of the constraint as we still prune all possible values. In addition, even when the Roots constraint is intractable to propagate completely, we can enforce bound consistency in linear time simply by enforcing bound consistency on the decomposition. Finally, we show that specifying counting and occurrence constraints using Roots is effective and efficient in practice on two benchmark problems from CSPLib.


Linear Time Integer Variable Global Constraint Global Nature Range 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beldiceanu, N.: Pruning for the minimum constraint family and for the number of distinct values constraint family. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 211–224. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. 2.
    Beldiceanu, N., Carlsson, M., Rampon, J.X.: Global constraint catalog. Technical Report T2005:08, SICS (2005)Google Scholar
  3. 3.
    Beldiceanu, N., Contejean, E.: Introducing global constraints in chip. Mathl. Comput. Modelling 20(12), 97–123 (1994)MATHCrossRefGoogle Scholar
  4. 4.
    Beldiceanu, N., Katriel, I., Thiel, S.: Filtering algorithms for the same and usedby constraints. MPI Technical Report MPI-I-2004-1-001 (2004)Google Scholar
  5. 5.
    Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: The Range and Roots constraints: Specifying counting and occurrence problems. In: Proc. of IJCAI 2005, pp. 60–65 (2005)Google Scholar
  6. 6.
    Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: Among, Common and Disjoint Constraints. LNCS (LNAI). Springer, Heidelberg (to appear) Google Scholar
  7. 7.
    Bessiere, C., Hebrard, E., Hnich, B., Kiziltan, Z., Walsh, T.: The Range constraint: Algorithms and implementation. In: Beck, J.C., Smith, B.M. (eds.) CPAIOR 2006. LNCS, vol. 3990. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Cheng, B.M.W., Choi, K.M.F., Lee, J.H.M., Wu, J.C.K.: Increasing constraint propagation by redundant modeling: an experience report. Constraints 4, 167–192 (1999)MATHCrossRefGoogle Scholar
  9. 9.
    Hnich, B., Kiziltan, Z., Walsh, T.: Modelling a Balanced Academic Curriculum Problem. In: Proc. of CPAIOR 2002, pp. 121–131 (2002)Google Scholar
  10. 10.
    Laburthe, F.: Choco: implementing a CP kernel. In: Proc. of CP 2000 Workshop TRICS: Techniques foR Implementing Constraint programming Systems (2000)Google Scholar
  11. 11.
    Quimper, C.-G., van Beek, P., López-Ortiz, A., Golynski, A., Sadjad, S.B.S.: An efficient bounds consistency algorithm for the global cardinality constraint. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 600–614. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    Refalo, P.: Linear formulation of constraint programming models and hybrid solvers. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 369–383. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Régin, J.C.: A filtering algorithm for constraints of difference in CSPs. In: Proc. of AAAI 1994, pp. 362–367 (1994)Google Scholar
  14. 14.
    Régin, J.C.: Generalized arc consistency for global cardinality constraint. In: Proc. of AAAI 1996, pp. 209–215 (1996)Google Scholar
  15. 15.
    Schulte, C., Stuckey, P.J.: Speeding up constraint propagation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 619–633. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Van Hentenryck, P., Deville, Y., Teng, C.M.: A generic arc-consistency algorithm and its specializations. Artificial Intelligence 57, 291–321 (1992)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christian Bessiere
    • 1
  • Emmanuel Hebrard
    • 2
  • Brahim Hnich
    • 3
  • Zeynep Kiziltan
    • 4
  • Toby Walsh
    • 5
  1. 1.LIRMMCNRS/University of MontpellierFrance
  2. 2.4C and UCCCorkIreland
  3. 3.Izmir University of EconomicsIzmirTurkey
  4. 4.University of BolognaItaly
  5. 5.NICTA and UNSWSydneyAustralia

Personalised recommendations