Pruning Rules for Constrained Optimisation for Conditional Preferences

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6876)


A depth-first search algorithm can be used to find optimal solutions of a Constraint Satisfaction Problem (CSP) with respect to a set of conditional preferences statements (e.g., a CP-net). This involves checking at each leaf node if the corresponding solution of the CSP is dominated by any of the optimal solutions found so far; if not, then we add this solution to the set of optimal solutions. This kind of algorithm can clearly be computationally expensive if the number of solutions is large. At a node N of the search tree, with associated assignment b to a subset of the variables B, it may happen that, for some previously found solution α, either (a) α dominates all extensions of b; or (b) α does not dominate any extension of a. The algorithm can be significantly improved if we can find sufficient conditions for (a) and (b) that can be efficiently checked. In case (a), we can backtrack since we need not continue the search below N; in case (b), α does not need to be considered in any node below the current node N. We derive a sufficient condition for (b), and three sufficient conditions for (a). Our experimental testing indicates that this can make a major difference to the efficiency of constrained optimisation for conditional preference theories including CP-nets.


Leaf Node Search Tree Constraint Satisfaction Problem Partial Assignment Conditional Preference 
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.
    Bessiere, C.: Constraint propagation. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming. Elsevier, Amsterdam (2006)Google Scholar
  2. 2.
    Bienvenu, M., Lang, J., Wilson, N.: From preference logics to preference languages, and back. In: Proc. KR 2010 (2010)Google Scholar
  3. 3.
    Boutilier, C., Brafman, R., Hoos, H., Poole, D.: Reasoning with conditional ceteris paribus preference statements. In: Proceedings of UAI 1999, pp. 71–80 (1999)Google Scholar
  4. 4.
    Boutilier, C., Brafman, R.I., Domshlak, C., Hoos, H., Poole, D.: CP-nets: A tool for reasoning with conditional ceteris paribus preference statements. Journal of Artificial Intelligence Research 21, 135–191 (2004)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Boutilier, C., Brafman, R.I., Domshlak, C., Hoos, H., Poole, D.: Preference-based constrained optimization with CP-nets. Computational Intelligence 20(2), 137–157 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brafman, R., Domshlak, C., Shimony, E.: On graphical modeling of preference and importance. Journal of Artificial Intelligence Research 25, 389–424 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Fahle, T., Schamberger, S., Sellmann, M.: Symmetry breaking. In: Walsh, T. (ed.) CP 2001. LNCS, vol. 2239, pp. 93–107. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  8. 8.
    Goldsmith, J., Lang, J., Truszczyński, M., Wilson, N.: The computational complexity of dominance and consistency in CP-nets. Journal of Artificial Intelligence Research 33, 403–432 (2008)MathSciNetzbMATHGoogle Scholar
  9. 9.
    McGeachie, M., Doyle, J.: Utility functions for ceteris paribus preferences. Computational Intelligence 20(2), 158–217 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Santhanam, G., Basu, S., Honavar, V.: Dominance testing via model checking. In: Proc. AAAI 2010 (2010)Google Scholar
  11. 11.
    Trabelsi, W., Wilson, N., Bridge, D., Ricci, F.: Comparing approaches to preference dominance for conversational recommender systems. In: Proc. ICTAI, pp. 113–118 (2010)Google Scholar
  12. 12.
    Wilson, N.: Consistency and constrained optimisation for conditional preferences. In: Proceedings of ECAI 2004, pp. 888–892 (2004)Google Scholar
  13. 13.
    Wilson, N.: Extending CP-nets with stronger conditional preference statements. In: Proceedings of AAAI 2004, pp. 735–741 (2004)Google Scholar
  14. 14.
    Wilson, N.: An efficient upper approximation for conditional preference. In: Proceedings of ECAI 2006, pp. 472–476 (2006)Google Scholar
  15. 15.
    Wilson, N.: Computational techniques for a simple theory of conditional preferences. Artificial Intelligence (in press, 2011), doi:10.1016/j.artint.2010.11.018Google Scholar
  16. 16.
    Wilson, N.: Efficient inference for expressive comparative preference languages. In: Proceedings of IJCAI 2009, pp. 961–966 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Cork Constraint Computation Centre, Department of Computer ScienceUniversity College CorkIreland

Personalised recommendations