A General Scheme for Multiple Lower Bound Computation in Constraint Optimization

  • Rina Dechter
  • Kalev Kask
  • Javier Larrosa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2239)


Computing lower bounds to the best-cost extension of a tuple is an ubiquous task in constraint optimization. A particular case of special interest is the computation of lower bounds to all singleton tuples, since it permits domain pruning in Branch and Bound algorithms. In this paper we introduce MCTE(z), a general algorithm which allows the computation of lower bounds to arbitrary sets of tasks. Its time and accuracy grows as a function of z allowing a controlled tradeo. between lower bound accuracy and time and space to fit available resources. Subsequently, a specialization of MCTE(z) called MBTE(z) is tailored to computing lower bounds to singleton tuples. Preliminary experiments on Max-CSP show that using MBTE(z) to guide dynamic variable and value orderings in branch and bound yields a dramatic reduction in the search space and, for some classes of problems, this reduction is highly coste effective producing significant time savings and is competitive against specialized algorithms for Max-CSP.


Constraint Satisfaction Soft Constraint Constraint Optimization Constraint Graph Lower Neighbor 
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.


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  1. [1]
    S.A. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability-a survey. BIT, 25:2–23, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    A. Becker and D. Geiger. A sufficiently fast algorithm for finding close to optimal junction trees. In Uncertainty in AI (UAI’96), pages 81–89, 1996.Google Scholar
  3. [3]
    C. Bessiere and J.-C. Regin. MAC and combined heuristics: Two reasons to forsake FC (and CBJ?) on hard problems. Lecture Notes in Computer Science, 1118:61–75, 1996.Google Scholar
  4. [4]
    S. Bistarelli, H. Fargier, U. Montanari, F. Rossi, T. Schiex, and G. Verfaillie. Semiring-based CSPs and valued CSPs: Frameworks, properties and comparison. Constraints, 4:199–240, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Bistarelly, R. Gennari, and F. Rossi. Constraint propagation for soft constraints: Generalization and termination conditions. In Proc. of the 6 th CP, pages 83–97, Singapore, 2000.Google Scholar
  6. [6]
    R. Debruyne and C. Bessière. Some practicable filtering techniques for the constraint satisfaction problem. In Proc. of the 16 th IJCAI, pages 412–417, Stockholm, Sweden, 1999.Google Scholar
  7. [7]
    R. Dechter. Bucket elimination: A unifying framework for reasoning. Artificial Intelligence, 113:41–85, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Dechter and J. Pearl. Tree clustering for constraint networks. Artificial Intelligence, 38:353–366, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Dechter and I. Rish. A scheme for approximating probabilistic inference. In Proceedings of the 13 th UAI-97, pages 132–141, San Francisco, 1997. Morgan Kaufmann Publishers.Google Scholar
  10. [10]
    R. Dechter and P. van Beek. Local and global relational consistency. Theoretical Computer Science, 173(1):283–308, 20 February 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    E. Freuder. A suficient condition for backtrack-free search. Journal of the ACM, 29:24–32, March 1982.Google Scholar
  12. [12]
    G. Gottlob, N. Leone, and F. Scarcello. A comparison of structural CSP decomposition methods. In Dean Thomas, editor, Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI-99-Vol1), pages 394–399, S.F., July 31-August 6 1999. Morgan Kaufmann Publishers.Google Scholar
  13. [13]
    K. Kask. new search heuristics for max-csp. In Proc. of the 6 th CP, pages 262–277, Singapore, 2000.Google Scholar
  14. [14]
    K. Kask and R. Dechter. A general scheme for automatic generation of search heuristics from specification dependencies. Artificial Intelligence, 129(1–2):91–131, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    K. Kask, J. Larrosa, and R. Dechter. A general scheme for multiple lower bound computation in constraint optimization. Technical report, University of California at Irvine, 2001.Google Scholar
  16. [16]
    J. Larrosa and P. Meseguer. Partial lazy forward checking for max-csp. In Proc. of the 13 th ECAI, pages 229–233, Brighton, United Kingdom, 1998.Google Scholar
  17. [17]
    J. Larrosa, P. Meseguer, and T. Schiex. Maintaining reversible DAC for max-CSP. Artificial Intelligence, 107(1):149–163, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    S. L. Lauritzen and D. J. Spiegelhalter. Local computation with probabilities on graphical structures and their applications to expert systems. Journal of the Royal Statistical Society, Series B, 34:157–224, 1988.MathSciNetGoogle Scholar
  19. [19]
    A. Mackworth. Consistency in networks of constraints. Artificial Intelligence, 8, 1977.Google Scholar
  20. [20]
    B. Nudel. Tree search and arc consistency in constraint satisfaction algorithms. Search in Artifical Intelligence, 999:287–342, 1988.Google Scholar
  21. [21]
    T. Schiex. Arc consistency for soft constraints. In Proc. of the 6 th CP, pages 411–424, Singapore, 2000.Google Scholar
  22. [22]
    P.P. Shenoy. Binary join-trees for computing marginals in the shenoy-shafer architecture. International Journal of Approximate Reasoning, 2–3:239–263, 1997.CrossRefMathSciNetGoogle Scholar
  23. [23]
    G. Verfaillie, M. Lemaître, and T. Schiex. Russian doll search. In Proc. of the 13 th AAAI, pages 181–187, Portland, OR, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Rina Dechter
    • 1
  • Kalev Kask
    • 1
  • Javier Larrosa
    • 2
  1. 1.University of California at Irvine (UCI)California
  2. 2.Universitat Politecnica de Catalunya (UPC)Catalunya

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