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An algebraic characterization of tractable constraints

  • Peter Jeavons
  • David Cohen
Session 11A: Scheduling
Part of the Lecture Notes in Computer Science book series (LNCS, volume 959)

Abstract

Many combinatorial search problems may be expressed as ‘constraint satisfaction problems’, and this class of problems is known to be NP-complete. In this paper we investigate what restrictions must be imposed on the allowed constraints in order to ensure tractability. We describe a simple algebraic closure condition, and show that this is both necessary and sufficient to ensure tractability in Boolean valued problems. We also demonstrate that this condition is necessary for problems with arbitrary finite domains.

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References

  1. 1.
    Cooper, M.C., Cohen, D.A., Jeavons, P.G., “Characterizing tractable constraints”, Artificial Intelligence 65, (1994), pp. 347–361.CrossRefGoogle Scholar
  2. 2.
    Cooper, M.C., & Jeavons, P.G., “Tractable constraints on ordered domains”, Technical Report, Dept of Computer Science, Royal Holloway, University of London, (1994) and submitted to Artificial Intelligence.Google Scholar
  3. 3.
    Csakany, B., “All minimal clones on the three-element set”, Acta Cybernetica 6, (1983), pp. 227–238.Google Scholar
  4. 4.
    Csakany, B., “On conservative minimal operations”, in Lectures in Universal Algebra (Proc. Conf. Szeged 1983), Colloq. Math. Soc. Janos Bolyai 43, North-Holland, (1986), pp. 49–60.Google Scholar
  5. 5.
    Dechter, R., & Pearl, J., “Structure identification in relational data”, Artificial Intelligence 58 (1992) pp. 237–270.CrossRefGoogle Scholar
  6. 6.
    Dechter, R. & Pearl J. “Network-based heuristics for constraint-satisfation problems”, Artificial Intelligence 34 (1988), pp. 1–38.CrossRefGoogle Scholar
  7. 7.
    Freuder, E. C. “A sufficient condition for backtrack-bounded search”, Journal of the ACM 32 (1985) pp. 755–761.CrossRefGoogle Scholar
  8. 8.
    Garey, M.R., & Johnson, D.S., Computers and intractability: a guide to NP-completeness, Freeman, San Francisco, California, (1979).Google Scholar
  9. 9.
    Geiger, D., “Closed systems of functions and predicates” Pacific Journal of Mathematics 27 (1968) pp. 95–100.Google Scholar
  10. 10.
    Gyssens, M., Jeavons, P., Cohen, D., “Decomposing constraint satisfaction problems using database techniques”, Artificial Intelligence 66, (1994), pp. 57–89.CrossRefGoogle Scholar
  11. 11.
    Jeavons, P.G., Cohen, D.A., Gyssens, M., “A unifying framework for tractable constraints”, Technical Report, Dept of Computer Science, Royal Holloway, University of London, (1995) to appear in Proceedings of Constraint Programming '95.Google Scholar
  12. 12.
    Kirousis, L., “Fast parallel constraint satisfaction”, Artificial Intelligence 64, (1993), pp. 147–160.CrossRefGoogle Scholar
  13. 13.
    Ladkin, P.B., & Maddux, R.D., “On binary constraint problems”, Journal of the ACM 41 (1994), pp. 435–469.CrossRefGoogle Scholar
  14. 14.
    Mackworth, A. K. “Consistency in networks of relations”, Artificial Intelligence 8 (1977) pp. 99–118.CrossRefGoogle Scholar
  15. 15.
    McKenzie, R.N., McNulty, G.F., Taylor, W.F., Algebras, lattices and varieties. Volume I, Wadsworth and Brooks, California (1987).Google Scholar
  16. 16.
    Montanari, U., “Networks of constraints: fundamental properties and applications to picture processing”, Information Sciences 7 (1974), pp. 95–132.CrossRefGoogle Scholar
  17. 17.
    Montanari, U., & Rossi, F., “Constraint relaxation may be perfect”, Artificial Intelligence 48 (1991), pp. 143–170.CrossRefGoogle Scholar
  18. 18.
    Post, E.L., “The two-valued iterative systems of mathematical logic”, Annals of Mathematical Studies 5, Princeton University Press, (1941).Google Scholar
  19. 19.
    Rosenberg, I.G., “Minimal clones I: the five types”, in Lectures in Universal Algebra (Proc. Conf. Szeged 1983), Colloq. Math. Soc. Janos Bolyai 43, North-Holland, (1986), pp. 405–427.Google Scholar
  20. 20.
    Schaefer, T.J., “The complexity of satisfiability problems”, Proc 10th ACM Symposium on Theory of Computing (STOC), (1978) pp. 216–226.Google Scholar
  21. 21.
    van Beek, P., “On the minimality and decomposability of row-convex constraint networks”, Proceedings of the Tenth National Conference on Artificial Intelligence, AAAI-92, MIT Press, (1992) pp. 447–452.Google Scholar
  22. 22.
    Van Hentenryck, P., Deville, Y., Teng, C-M., “A generic arc-consistency algorithm and its specializations”, Artificial Intelligence 57 (1992), pp. 291–321.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Peter Jeavons
    • 1
  • David Cohen
    • 1
  1. 1.Department of Computer Science, Royal HollowayUniversity of LondonUK

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