This paper shows that existing definitions of costs associated with soft global constraints are not sufficient to deal with all the usual global constraints. We propose more expressive definitions: refined variable-based cost, object-based cost and graph properties based cost. For the first two ones we provide ad-hoc algorithms to compute the cost from a complete assignment of values to variables. A representative set of global constraints is investigated. Such algorithms are generally not straightforward and some of them are even NP-Hard. Then we present the major feature of the graph properties based cost: a systematic way for evaluating the cost with a polynomial complexity.


Constraint Satisfaction Problem Cost Evaluation Global Constraint Graph Property Polynomial Complexity 
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.
    Aggoun, A., Beldiceanu, N.: Extending CHIP in order to solve complex scheduling and placement problems. Mathl. Comput. Modelling 17(7), 57–73 (1993)CrossRefGoogle Scholar
  2. 2.
    Ahuja, R.K., Orlin, J.B., Stein, C., Tarjan, R.E.: Improved algorithms for bipartite network flow. SIAM Journal on Computing 23(5), 906–933 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bajard, J.-C., Common, H., Kenion, C., Krob, D., Muller, J.-M., Petit, A., Robert, Y., Morvan, M.: Exercices d’algorithmique: oraux d’ENS. In: International Thomson Publishing, pp. 72–74 (1997) (in French)Google Scholar
  4. 4.
    Baptiste, P., Le Pape, C., Peridy, L.: Global constraints for partial CSPs: A case-study of resource and due date constraints. In: Maher, M.J., Puget, J.-F. (eds.) CP 1998. LNCS, vol. 1520, pp. 87–102. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Beldiceanu, N.: Global constraints as graph properties on a structured network of elementary constraints of the same type. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 52–66. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Beldiceanu, N., Carlsson, M.: Revisiting the cardinality operator and introducing the cardinality-path constraint family. In: Codognet, P. (ed.) ICLP 2001. LNCS, vol. 2237, pp. 59–73. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Beldiceanu, N., Contejean, E.: Introducing global constraints in CHIP. Journal of Mathematical and Computer Modelling 20(12), 97–123 (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Berge, C.: Graphs and hypergraphs. Dunod, Paris (1970)Google Scholar
  10. 10.
    Bessière, C., Van Hentenryck, P.: To be or not to be.. a global constraint. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 789–794. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Bistarelli, S., Montanari, U., Rossi, F., Schiex, T., Verfaillie, G., Fargier, H.: Semiring-based CSPs and valued CSPs: Frameworks, properties, and comparison. Constraints 4, 199–240 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Dechter, R.: Constraint networks. Encyclopedia of Artificial Intelligence, 276–285 (1992)Google Scholar
  13. 13.
    Dechter, R., Pearl, J.: Network-based heuristics for constraint-satisfaction problems. Artificial Intelligence 34, 1–38 (1987)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Ford, L., Flukerson, D.: Flows in networks. Princeton University Press, Princeton (1962)zbMATHGoogle Scholar
  15. 15.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are np-complete. Inform. Processing Letters 12(3), 133–137 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. W.H. Freeman and Company, New York (1979) ISBN 0-7167-1045-5zbMATHGoogle Scholar
  17. 17.
    Gent, I., Stergiou, K., Walsh, T.: Decomposable constraints. Artificial Intelligence 123, 133–156 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Lawler, E.: Combinatorial optimization: Networks and matroids. Holt, Rinehart and Winston (1976)zbMATHGoogle Scholar
  19. 19.
    Pachet, F., Roy, P.: Automatic generation of music programs. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 331–345. Springer, Heidelberg (1999)Google Scholar
  20. 20.
    Petit, T., Régin, J.-C., Bessière, C.: Meta constraints on violations for over constrained problems. In: Proceedings IEEE-ICTAI, pp. 358–365 (2000)Google Scholar
  21. 21.
    Petit, T., Régin, J.-C., Bessière, C.: Specific filtering algorithms for over constrained problems. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 451–463. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Régin, J.-C.: A filtering algorithm for constraints of difference in CSPs. In: Proceedings AAAI, pp. 362–367 (1994)Google Scholar
  23. 23.
    Régin, J.-C.: Generalized arc consistency for global cardinality constraint. In: Proceedings AAAI, pp. 209–215 (1996)Google Scholar
  24. 24.
    Van Hentenryck, P., Michel, L.: Control abstractions for local search. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 66–80. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Nicolas Beldiceanu
    • 1
  • Thierry Petit
    • 1
  1. 1.LINA FRE CNRS 2729 École des Mines de NantesNantes Cedex 3France

Personalised recommendations