Mini-bucket Elimination with Bucket Propagation

  • Emma Rollon
  • Javier Larrosa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4204)


Many important combinatorial optimization problems can be expressed as constraint satisfaction problems with soft constraints. When problems are too difficult to be solved exactly, approximation methods become the best option. Mini-bucket Elimination (MBE) is a well known approximation method for combinatorial optimization problems. It has a control parameter z that allow us to trade time and space for accuracy. In practice, it is the space and not the time that limits the execution with high values of z. In this paper we introduce a new propagation phase that MBE should execute at each bucket. The purpose of this propagation is to jointly process as much information as possible. As a consequence, the undesirable lose of accuracy caused by MBE when splitting functions into different mini-buckets is minimized. We demonstrate our approach in scheduling, combinatorial auction and max-clique problems, where the resulting algorithm MBE p gives important percentage increments of the lower bound (typically 50% and up to 1566%) with only doubling the cpu time.


Propagation Tree Space Complexity Propagation Phase Constraint Satisfaction Problem Soft 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.
    Dechter, R., Rish, I.: Mini-buckets: A general scheme for bounded inference. Journal of the ACM 50, 107–153 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bistarelli, S., Fargier, H., Montanari, U., Rossi, F., Schiex, T., Verfaillie, G.: Semiring-based CSPs and valued CSPs: Frameworks, properties and comparison. Constraints 4, 199–240 (1999)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Pearl, J.: Probabilistic Inference in Intelligent Systems. In: Pearl, J. (ed.) Networks of Plausible Inference. Morgan Kaufmann, San Mateo (1988)Google Scholar
  4. 4.
    Xu, H., Rutenbar, R.A., Sakallah, K.: sub-sat: A formulation for relaxed boolean satisfiability with applications in rounting. In: Proc. Int. Symp. on Physical Design, CA (2002)Google Scholar
  5. 5.
    Strickland, D.M., Barnes, E., Sokol, J.S.: Optimal protein structure alignment using maximum cliques. Operations Research 53, 389–402 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Vasquez, M., Hao, J.: A logic-constrained knapsack formulation and a tabu algorithm for the daily photograph scheduling of an earth observation satellite. Journal of Computational Optimization and Applications 20(2) (2001)Google Scholar
  7. 7.
    Park, J.D.: Using weighted max-sat engines to solve mpe. In: Proc. of the 18th AAAI, Edmonton, Alberta, Canada, pp. 682–687 (2002)Google Scholar
  8. 8.
    Kask, K., Dechter, R.: A general scheme for automatic generation of search heuristics from specification dependencies. Artificial Intelligence 129, 91–131 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Rollon, E., Larrosa, J.: Depth-first mini-bucket elimination. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 563–577. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Cooper, M., Schiex, T.: Arc consistency for soft constraints. Artificial Intelligence 154, 199–227 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dechter, R.: Bucket elimination: A unifying framework for reasoning. Artificial Intelligence 113, 41–85 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bertele, U., Brioschi, F.: Nonserial Dynamic Programming. Academic Press, London (1972)MATHGoogle Scholar
  13. 13.
    Larrosa, J., Schiex, T.: Solving weighted csp by maintaining arc-consistency. Artificial Intelligence 159, 1–26 (2004)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Cooper, M.: High-order consistency in valued constraint satisfaction. Constraints 10, 283–305 (2005)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bensana, E., Lemaitre, M., Verfaillie, G.: Earth observation satellite management. Constraints 4(3), 293–299 (1999)MATHCrossRefGoogle Scholar
  16. 16.
    Leuton-Brown, Y.S.K., Pearson, M.: Towards a universal test suite for combinatorial auction algorithms. ACM E-Commerce, 66–76 (2000)Google Scholar
  17. 17.
    Johnson, D.S., Trick, M.: Second dimacs implementation challenge: cliques, coloring and satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science. AMS, vol. 26 (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Emma Rollon
    • 1
  • Javier Larrosa
    • 1
  1. 1.Universitat Politecnica de CatalunyaBarcelonaSpain

Personalised recommendations