A Totally Unimodular Description of the Consistent Value Polytope for Binary Constraint Programming

  • Ionuţ D. Aron
  • Daniel H. Leventhal
  • Meinolf Sellmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3990)


We present a theoretical study on the idea of using mathematical programming relaxations for filtering binary constraint satisfaction problems. We introduce the consistent value polytope and give a linear programming description that is provably tighter than a recently studied formulation. We then provide an experimental study that shows that, despite the theoretical progress, in practice filtering based on mathematical programming relaxations continues to perform worse than standard arc-consistency algorithms for binary constraint satisfaction problems.


Cost-based filtering hybrid methods mathematical programming 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahuja, R.K., Magnati, T.L., Orlin, J.B.: Network Flows. Prentice Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Appa, G., Magos, D., Mourtos, I.: An LP-based proof for the non-existence of a pair of Orthogonal Latin Squares for n=6. OR Letters 32(4), 336–344 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bessiere, C.: Random Uniform CSP Generators,
  4. 4.
    Fahle, T., Junker, U., Karisch, S.E., Kohl, N., Sellmann, M., Vaaben, B.: Constraint programming based column generation for crew assignment. Journal of Heuristics 8(1), 59–81 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fahle, T., Sellmann, M.: Cost-Based Filtering for the Constrained Knapsack Problem. Annals of Operations Research 115, 73–93 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Focacci, F., Lodi, A., Milano, M.: Cutting Planes in Constraint Programming: An Hybrid Approach. In: Proceedings of CP-AI-OR 2000, Paderborn Center for Parallel Computing, Technical Report tr-001-2000, pp. 45–51 (2000)Google Scholar
  7. 7.
    Focacci, F., Lodi, A., Milano, M.: Cost-Based Domain Filtering. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 189–203. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Hooker, J.N.: A hybrid method for planning and scheduling. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 305–316. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Hooker, J.N., Ottosson, G.: Logic-based Benders decomposition. Mathematical Programming 96, 33–60 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    ILOG SA. ILOG Concert 2.0,
  11. 11.
    Junker, U., Karisch, S.E., Kohl, N., Vaaben, B., Fahle, T., Sellmann, M.: A Framework for Constraint programming based column generation. In: Jaffar, J. (ed.) CP 1999. LNCS, vol. 1713, pp. 261–275. Springer, Heidelberg (1999)Google Scholar
  12. 12.
    Khemmoudj, M.O.I., Bennaceur, H., Nagih, A.: Combining Arc-Consistency and Dual Lagrangean Relaxation for Filtering CSPs. In: Barták, R., Milano, M. (eds.) CPAIOR 2005. LNCS, vol. 3524, pp. 258–272. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Kim, H.-J., Hooker, J.N.: Solving fixed-charge network flow problems with a hybrid optimization and constraint programming approach. Annals of Operations Research 115, 95–124 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Milano, M.: Integration of Mathematical Programming and Constraint Programming for Combinatorial Optimization Problems. In: Tutorial at CP 2000 (2000)Google Scholar
  15. 15.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, Chichester (1988)CrossRefzbMATHGoogle Scholar
  16. 16.
    Ottosson, G., Thorsteinsson, E.S.: Linear Relaxation and Reduced-Cost Based Propagation of Continuous Variable Subscripts. In: CP-AI-OR 2000, Paderborn Center for Parallel Computing, Technical Report tr-001-2000, pp. 129–138 (2000)Google Scholar
  17. 17.
    Régin, J.-C.: Cost-Based Arc Consistency for Global Cardinality Constraints. Constraints 7(3-4), 387–405 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sellmann, M.: Theoretical Foundations of CP-based Lagrangian Relaxation. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 634–647. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Sellmann, M.: Approximated Consistency for Knapsack Constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 679–693. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Sellmann, M., Fahle, T.: Constraint Programming Based Lagrangian Relaxation for the Automatic Recording Problem. Annals of Operations Research 118, 17–33 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sellmann, M., Fahle, T.: Coupling Variable Fixing Algorithms for the Automatic Recording Problem. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 134–145. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Sellmann, M., Harvey, W.: Heuristic Constraint Propagation. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 738–743. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ionuţ D. Aron
    • 1
  • Daniel H. Leventhal
    • 1
  • Meinolf Sellmann
    • 1
  1. 1.Department of Computer ScienceBrown UniversityProvidenceU.S.A.

Personalised recommendations