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A new tractable class of constraint satisfaction problems

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Abstract

In this paper we consider constraint satisfaction problems where the set of constraint relations is fixed. Feder and Vardi (1998) identified three families of constraint satisfaction problems containing all known polynomially solvable problems. We introduce a new class of problems called para-primal problems, incomparable with the families identified by Feder and Vardi (1998) and we prove that any constraint problem in this class is decidable in polynomial time. As an application of this result we prove a complete classification for the complexity of constraint satisfaction problems under the assumption that the basis contains all the permutation relations. In the proofs, we make an intensive use of algebraic results from clone theory about the structure of para-primal and homogeneous algebras.

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References

  1. A. Bulatov, Combinatorial problems raised from semilattices, Manuscript.

  2. A. Bulatov, A dichotomy theorem for constraints on a three-element set, in: Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science (FOCS’02) (2002) pp. 649–658.

  3. A. Bulatov, Malt’sev constraints are tractable, Technical Report PRG-RR-02-05, Computing Laboratory, Oxford University (2002).

  4. A. Bulatov, Tractable conservative constraint satisfaction problems, in: Proceedings of 18th IEEE Symposium on Logic in Computer Science (LICS’03) (2003) pp. 321–330.

  5. A. Bulatov, P. Jeavons and M. Volkov, Finite semigroups imposing tractable constraints, in: Proceedings of the School on Algorithmic Aspects of the Theory of Semigroups and its Applications (2002) pp. 313–329.

  6. A. Bulatov and P.G. Jeavons, Tractable constraints closed under a binary operation, Technical Report PRG-TR-12-00, Computing Laboratory, Oxford University (2000).

  7. A. Bulatov, A. Krokhin and P. Jeavons, Constraint satisfaction problems and finite algebras, Technical Report PRG-TR-4-99, Computing Laboratory, Oxford University (1999).

  8. D.M. Clark and P.H. Krauss, Para primal algebras, Algebra Universalis 6 (1976) 165–192.

    Google Scholar 

  9. D.M. Clark and P.H. Krauss, Plain para primal algebras, Algebra Universalis 11 (1980) 365–388.

    Google Scholar 

  10. S.A. Cook, The complexity of theorem-proving procedures, in: 3rd Annual ACM Symposium on Theory of Computing (STOC’71) (1971) pp. 151–158.

  11. M.C. Cooper, D.A. Cohen and P.G. Jeavons, Characterizing tractable constraints, Artificial Intelligence 65 (1994) 347–361.

    Google Scholar 

  12. V. Dalmau and P. Jeavons, Learnability of quantified formulas, in: 4th European Conference on Computational Learning Theory Eurocolt’99, Lecture Notes in Artificial Intelligence, Vol. 1572 (Springer, Berlin, 1999) pp. 63–78.

    Google Scholar 

  13. V. Dalmau and J. Pearson, Set functions and width 1, in: 5th International Conference on Principles and Practice of Constraint Programming, CP’99, Lecture Notes in Computer Science, Vol. 1713 (Springer, Berlin, 1999) pp. 159–173.

    Google Scholar 

  14. R. Dechter and J. Pearl, Network-based heuristics for constraint satisfaction problems, Artificial Intelligence 34(1) (1988) 1–38.

    Google Scholar 

  15. T. Feder and M.Y. Vardi, The computational structure of monotone monadic SNP and contraint satisfaction: A study through datalog and group theory, SIAM J. Computing 28(1) (1998) 57–104.

    Google Scholar 

  16. E.C. Freuder, A sufficient condition for backtrack-bounded search, J. ACM 32 (1985) 755–761.

    Google Scholar 

  17. M. Gyssens, P. Jeavons and D. Cohen, Decomposing constraint satisfaction problems using database techniques, Artificial Intelligence 66(1) (1994) 57–89.

    Google Scholar 

  18. P. Hell and J. Ne \u{s} et \u{r} il, On the complexity of H-coloring, J. Comb. Theory, Series B 48 (1990) 92–110.

  19. P. Jeavons, On the algebraic structure of combinatorial problems, Theoretical Computer Science 200 (1998) 185–204.

    Google Scholar 

  20. P. Jeavons, D. Cohen and M.C. Cooper, Constraints, consistency and closure, Artificial Intelligence 101 (1998) 251–265.

    Google Scholar 

  21. P. Jeavons, D. Cohen and M. Gyssens, A unifying framework for tractable constraints, in: 1st International Conference on Principles and Practice of Constraint Programming, CP’95, Cassis (France), September 1995, Lecture Notes in Computer Science, Vol. 976 (Springer, Berlin, 1995) pp. 276–291.

    Google Scholar 

  22. P. Jeavons, D. Cohen and M. Gyssens, A test for tractability, in: 2nd International Conference on Principles and Practice of Constraint Programming CP’96, Lecture Notes in Computer Science, Vol. 1118 (Springer, Berlin, 1996) pp. 267–281.

    Google Scholar 

  23. P. Jeavons, D. Cohen and M. Gyssens, Closure properties of constraints, J. ACM 44(4) (1997) 527–548.

    Google Scholar 

  24. P. Jeavons and M. Cooper, Tractable constraints on ordered domains, Artificial Intelligence 79 (1996) 327–339.

    Google Scholar 

  25. L. Kirousis, Fast parallel constraint satisfaction, Artificial Intelligence 64 (1993) 147–160.

    Google Scholar 

  26. A.K. Mackworth, Consistency in networks of relations, Artificial Intelligence 8 (1977) 99–118.

    Google Scholar 

  27. S.S. Marchenkov, Homogeneous algebras, Problemy Kibernetiki 39 (1982) 85–106.

    Google Scholar 

  28. E. Marczewski, Homogeneous algebras and homogeneous operations, Fund. Math. 56 (1964) 81–103.

    Google Scholar 

  29. R.N. McKenzie, G.F. McNulty and W.F. Taylor, Algebras, Lattices and Varieties, Vol. 1 (Wadsworth and Brooks, 1987).

  30. U. Montanari, Networks of constraints: Fundamental properties and applications to picture processing, Information Sciences 7 (1974) 95–132.

    Google Scholar 

  31. U. Montanari and F. Rossi, Constraint relaxation may be perfect, Artificial Intelligence 48 (1991) 143–170.

    Google Scholar 

  32. R.W. Quackenbush, Minimal para primal algebras, in: Contributions to General Algebra 2, Proc. Klagenfurt Conf. 1982 (1983) pp. 291–304.

  33. T.J. Schaefer, The complexity of satisfiability problems, in: 10th Annual ACM Symposium on Theory of Computing (1978) pp. 216–226.

  34. A. Szendrei, Clones in Universal Algebra, Seminaires de Mathématiques Supéreiores, Vol. 99 (University of Montreal, Montreal, 1986).

    Google Scholar 

  35. P. van Beek and R. Dechter, On the minimality and decomposability of row-convex constraint networks, J. ACM 42 (1995) 543–561.

    Google Scholar 

  36. P. van Hentenryck, Y. Deville and C.-M. Teng, A generic Arc-consistency algorithm and its specializations, Artificial Intelligence (1992).

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  1. Departament de Tecnologia, Universitat Pompeu Fabra Estació de França, Passeig de la circumval.lació, Barcelona, 08003, Spain

    Víctor Dalmau

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  1. Víctor Dalmau
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Dalmau, V. A new tractable class of constraint satisfaction problems. Ann Math Artif Intell 44, 61–85 (2005). https://doi.org/10.1007/s10472-005-1810-9

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