Network-Based Heuristics for Constraint-Satisfaction Problems

  • Rina Dechter
  • Judea Pearl
Part of the Symbolic Computation book series (SYMBOLIC)

Abstract

Many AI tasks can be formulated as Constraint-Satisfaction problems (CSP), i.e., the assignment of values to variables subject to a set of constraints. While some CSPs are hard, those that are easy can often be mapped into sparse networks of constraints which, in the extreme case, are trees. This paper identifies classes of problems that lend themselves to easy solutions, and develops algorithms that solve these problems optimally. The paper then presents a method of generating heuristic advice to guide the order of value assignments based on both the sparseness found in the constraint network and the simplicity of tree-structured CSPs. The advice is generated by simplifying the pending subproblems into trees, counting the number of consistent solutions in each simplified subproblem, and comparing these counts to decide among the choices pending in the original problem.

Keywords

Expense Doyle Brioschi 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Arnborg,1985]
    S. Amborg, “Efficient algorithms for combinatorial problems on graphs with bounded decomposability–a survey,” BIT, Vol. 25, 1985, pp. 2–23.MathSciNetGoogle Scholar
  2. [Arnborg,1987]
    S. Amborg, D. G. Cornell, and A. Proskurowski, “Complexity of finding embeddings in a k-tree,” Siam Journal of algorithm and Discrete Math.,Vol. 8, No. 2, 1987., pp. 277–184.CrossRefGoogle Scholar
  3. [Bertele,1972]
    U. Bertele and F. Brioschi, Nonserial Dynamic Programming, New York: Academic press, 1972.MATHGoogle Scholar
  4. [Bruynooghe,1984]
    Maurice Bruynooghe and Luis M. Pereira, “Deduction Revision by Intelligent backtracking,” in Implementation of Prolog, J.A. Campbell, Ed. Ellis Harwood, 1984, pp. 194–215.Google Scholar
  5. [Carbone11,1983]
    J.G. Carbonell, “Learning by analogy: Formulation and generating plan from past experience,” in Machine Learning, Michalski, Carbonell and Mitchell, Ed. Palo Alto, California: Tioga Press, 1983.Google Scholar
  6. [Cox,1984]
    P.T. Cox, “Finding backtrack points for intelligent backtracking,” in Implementation of Prolog, J.A. Campbell, Ed. Ellis Harwood, 1984, pp. 216–233.Google Scholar
  7. [Dechter,1985a]
    R. Dechter and J.Pearl, “A problem simplification approach that generates heuristics for constraint satisfaction problems.,” UCLA-Eng-rep.8497. To appear in Machine Intelligence 11., 1985.Google Scholar
  8. [Dechter,1985b]
    R. Dechter and J. Pearl, “The anatomy of easy problems: a constraint-satisfaction formulation,” in Proceedings Ninth International Conference on Artificial Intelligence, Los Angeles, Cal: 1985, pp. 1066–1072.Google Scholar
  9. [Dechter,1986]
    R. Dechter, “Learning while searching in constraintsatisfaction-problems,” in Proceedings AAAI-86, Philadelphia, Pensilvenia: 1986.Google Scholar
  10. [Dechter,1987a]
    A. Dechter and R. Dechter, “Removing redundencies in constraint networks,” in Proceedings AAAI-87, Seattle, Washington: 1987Google Scholar
  11. [Dechter,1987b]
    Rina Dechter, “Tree-Clustering Schemes for Constraint-Processing,” Cognitive Systems Laboratory, Tech. Rep. (R-92), 1987.Google Scholar
  12. [Dechter,1987c]
    R. Dechter and J. Pearl, “The cycle-cutset method for improving search performance in AI applications,” in Proceeding of the 3rd IEEE on AI Applications, Orlando, Florida: 1987.Google Scholar
  13. [Doyle,1979]
    John Doyle, “A truth maintenance system,” Artificial Intelligence, Vol. 12, 1979, pp. 231–272.MathSciNetCrossRefGoogle Scholar
  14. [Even,1979]
    S. Even, Graph Algorithms, Maryland, USA: Computer Science Press, 1979.MATHGoogle Scholar
  15. [Freuder,1982]
    E.C. Freuder, “A sufficient condition of backtrack-free search.,” Journal of the ACM, Vol. 29, No. 1, 1982, pp. 24–32.MathSciNetMATHCrossRefGoogle Scholar
  16. [Freuder,1985]
    E.C. Freuder, “A sufficient condition for backtrack-bounded search,” Journal of the Association of Computing Machinery, Vol. 32, No. 4, 1985, pp. 755–761.MathSciNetMATHCrossRefGoogle Scholar
  17. [Gaschnig,1979a]
    J. Gaschnig, “Performance measurement and analysis of certain search algorithms.,” Carnegie-Mellon University, Pitsburg, Pensilvenia, Tech. Rep. CMU-CS79–124, 1979.Google Scholar
  18. [Gaschnig,1979b]
    J. Gaschnig, “A problem similarity approach to devising heuristics: first results,” in Proceedings 6th international joint conf. on Artificial Intelligence., Tokyo, Jappan: 1979, pp. 301–307.Google Scholar
  19. [Guida,1979]
    G. Guida and M. Somalvico, “A method for computing heuristics in problem solving,” Information Sciences, Vol. 19, 1979, pp. 251–259.MathSciNetMATHCrossRefGoogle Scholar
  20. [Haralick,1980]
    R. M. Haralick and G.L. Elliot, “Increasing tree search efficiency for constraint satisfaction problems,” AI Journal, Vol. 14, 1980, pp. 263–313.Google Scholar
  21. [Knuth,1975]
    D. E. Knuth, “Esimating the efficiency of backtrack programs,” Mathematics of computation, Vol. 29, No. 129, 1975, pp. 121–136.MathSciNetMATHCrossRefGoogle Scholar
  22. [Mackworth,1977]
    A.K. Mackworth, “Consistency in networks of relations,” Artifficial intelligence, Vol. 8, No. 1, 1977, pp. 99–118.MATHCrossRefGoogle Scholar
  23. [Mackworth,1984]
    A.K. Mackworth and E.C. Freuder, “The complexity of some polynomial network consistancy algorithms for constraint satisfaction problems,” Artificial Intelligence, Vol. 25, No. 1, 1984.Google Scholar
  24. [Martins,1986]
    Joao P. Martins and Stuart C. Shapiro, “Theoretical Foundations for belief revision,” in Proceedings Theoretical aspects of Reasoning about knowledge, 1986.Google Scholar
  25. [Matwin,1985]
    Stanislaw Matwin and Tomasz Pietrzykowski, “Intelligent backtracking in plan-based deduction,” IEEE Transaction on Pattern Analysis and Machine Intelligence,Vol. PAMI-7, No. 6, 1985, pp. 682–692.MATHCrossRefGoogle Scholar
  26. [Minsky,1963]
    M. Minsky, “Steps towards Artificial Intelligence,” in Computer and Thought, Feigenbaoum Feldman, Ed. McGraw-Hill, 1963, p. 442.Google Scholar
  27. [Mohr,1986]
    R. Mohr and T.C. Henderson, “Arc and Path consistency revisited,” Artificial Intelligence, Vol. 28, No. 2, 1986, pp. 225–233.CrossRefGoogle Scholar
  28. [Montanari,1974]
    U. Montanari, “Networks of constraints:fundamental properties and applications to picture processing,” Information Science, Vol. 7, 1974, pp. 95–132.MathSciNetCrossRefGoogle Scholar
  29. [Nudel,1983]
    B. Nudel, “Consistent-Labeling problems and their algorithms. Expected complexities and theory based heuristics.,” Artificial Intelligence, Vol. 21, 1983, pp. 135–178.CrossRefGoogle Scholar
  30. [Pear1,1983]
    J. Pearl, “On the discovery and generation of certain heuristics,” AI Magazine, No. 22–23, 1983.Google Scholar
  31. [Purdom,1983]
    P. Purdom, “Search rearrangement backtracking and polynomial average time,” AI Journal, Vol. 21, 1983, pp. 117–133.Google Scholar
  32. [Purdom,1985]
    P.W. Purdom and C.A. Brown, The Analysis of Algorithms: CBS College Publishing, Holt, Rinehart and Winston, 1985.Google Scholar
  33. [Sacerdoti,1974]
    E. D. Sacerdoti, “Planning in a hierarchy of abstraction spaces,” Artificial Intelligence, Vol. 5, No. 2, 1974, pp. 115–135.MATHCrossRefGoogle Scholar
  34. [Stallman,1977]
    R.M. Stallman and G. J. Sussman, “Forward reasoning and dependency-directed backtracking in a system for computer-aided circuit analysis,” Artificial Intelligence, Vol. 9, No. 2, 1977, pp. 135–196.MATHCrossRefGoogle Scholar
  35. [Stone,1986a]
    H. S. Stone and J. M. Stone, “Efficient search techniques- An empirical study of the N-queens problem.,” IBM T.J. Watson Research Center, Yorktown Heights,NY Tech. Rep. RC12057 (54343) 1986.Google Scholar
  36. [Stone,1986b]
    H.S. Stone and P. Sipala, “The average complexity of depth-first search with backtracking and cut-off,” IBM JRD, Vol. 30, No. 3, 1986, pp. 242–258.MATHCrossRefGoogle Scholar
  37. [Tarjan,1984]
    R. Tarjan and M. Yannakakis, “Simple Linear-Time Alsorithms to test Chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs,” SIAM Journal of Computing, Vol. 13, No. 3, 1984, pp. 566–579.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Rina Dechter
    • 1
    • 2
  • Judea Pearl
    • 1
  1. 1.Cognitive Systems Laboratory, Computer Science DepartmentUniversity of CaliforniaLos AngelesUSA
  2. 2.Artificial Intelligence CenterHughes Aircraft CompanyCalabasasUSA

Personalised recommendations