Filtering Numerical CSPs Using Well-Constrained Subsystems

  • Ignacio Araya
  • Gilles Trombettoni
  • Bertrand Neveu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5732)


When interval methods handle systems of equations over the reals, two main types of filtering/contraction algorithms are used to reduce the search space. When the system is well-constrained, interval Newton algorithms behave like a global constraint over the whole n ×n system. Also, filtering algorithms issued from constraint programming perform an AC3-like propagation loop, where the constraints are iteratively handled one by one by a revise procedure. Applying a revise procedure amounts in contracting a 1 ×1 subsystem.

This paper investigates the possibility of defining contracting well-constrained subsystems of size k (1 ≤ k ≤ n). We theoretically define the Box-k-consistency as a generalization of the state-of-the-art Box-consistency. Well-constrained subsystems act as global constraints that can bring additional filtering w.r.t. interval Newton and 1 ×1 standard subsystems. Also, the filtering performed inside a subsystem allows the solving process to learn interesting multi-dimensional branching points, i.e., to bisect several variable domains simultaneously. Experiments highlight gains in CPU time w.r.t. state-of-the-art algorithms on decomposed and structured systems.


Local Tree Search Tree Constraint Programming Variable Domain Global 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.
    Ait-Aoudia, S., Jegou, R., Michelucci, D.: Reduction of Constraint Systems. In: Compugraphic (1993)Google Scholar
  2. 2.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising Hull and Box Consistency. In: Proc. ICLP, pp. 230–244 (1999)Google Scholar
  3. 3.
    Chabert, G.: (2009)
  4. 4.
    Chabert, G., Jaulin, L.: Contractor Programming. Artificial Intelligence 173, 1079–1100 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cruz, J., Barahona, P.: Global Hull Consistency with Local Search for Continuous Constraint Solving. In: Brazdil, P.B., Jorge, A.M. (eds.) EPIA 2001. LNCS (LNAI), vol. 2258, pp. 349–362. Springer, Heidelberg (2001)Google Scholar
  6. 6.
    Debruyne, R., Bessière, C.: Some Practicable Filtering Techniques for the Constraint Satisfaction Problem. In: Proc. IJCAI, pp. 412–417 (1997)Google Scholar
  7. 7.
    Dulmage, A.L., Mendelsohn, N.S.: Covering of Bipartite Graphs. Canadian Journal of Mathematics 10, 517–534 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kearfott, R.B., Novoa III, M.: INTBIS, a portable interval Newton/Bisection package. ACM Trans. on Mathematical Software 16(2), 152–157 (1990)CrossRefzbMATHGoogle Scholar
  10. 10.
    Lebbah, Y., Michel, C., Rueher, M., Daney, D., Merlet, J.P.: Efficient and safe global constraints for handling numerical constraint systems. SIAM Journal on Numerical Analysis 42(5), 2076–2097 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lhomme, O.: Consistency Tech. for Numeric CSPs. In: IJCAI, pp. 232–238 (1993)Google Scholar
  12. 12.
  13. 13.
    Neumaier, A.: Int. Meth. for Systems of Equations. Cambridge Univ. Press, Cambridge (1990)Google Scholar
  14. 14.
    Neveu, B., Chabert, G., Trombettoni, G.: When Interval Analysis helps Interblock Backtracking. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 390–405. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Neveu, B., Jermann, C., Trombettoni, G.: Inter-Block Backtracking: Exploiting the Structure in Continuous CSPs. In: Jermann, C., Neumaier, A., Sam, D. (eds.) COCOS 2003. LNCS, vol. 3478, pp. 15–30. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  16. 16.
    Régin, J.-C.: A Filtering Algorithm for Constraints of Difference in CSPs. In: Proc. AAAI 1994, pp. 362–367 (1994)Google Scholar
  17. 17.
    Trombettoni, G., Chabert, G.: Constructive Interval Disjunction. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 635–650. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ignacio Araya
    • 1
  • Gilles Trombettoni
    • 1
  • Bertrand Neveu
    • 1
  1. 1.INRIAUniversité de Nice-Sophia, CERTISFrance

Personalised recommendations