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Abstract

We study the computational complexity of binary valued constraint satisfaction problems (VCSP) given by allowing only certain types of costs in every triangle of variable-value assignments to three distinct variables. We show that for several computational problems, including CSP, Max-CSP, finite-valued VCSP, and general-valued VCSP, the only non-trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property [9].

Keywords

Constraint Satisfaction Problem Cost Type Tractable Case Constraint Satisfaction Problem Instance Unary Cost 
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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References

  1. 1.
    Bertelé, U., Brioshi, F.: Nonserial dynamic programming. Academic Press, London (1972)zbMATHGoogle Scholar
  2. 2.
    Bistarelli, S., Montanari, U., Rossi, F.: Semiring-based Constraint Satisfaction and Optimisation. Journal of the ACM 44(2), 201–236 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bulatov, A., Krokhin, A., Jeavons, P.: Classifying the Complexity of Constraints using Finite Algebras. SIAM Journal on Computing 34(3), 720–742 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cohen, D.A., Cooper, M.C., Jeavons, P.G.: Generalising submodularity and Horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science 401(1-3), 36–51 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohen, D.A., Cooper, M.C., Jeavons, P.G., Krokhin, A.A.: The Complexity of Soft Constraint Satisfaction. Artificial Intelligence 170(11), 983–1016 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cohen, D., Jeavons, P.: The complexity of constraint languages. In: Rossi, F., van Beek, P., Walsh, T. (eds.) The Handbook of Constraint Programming. Elsevier, Amsterdam (2006)Google Scholar
  7. 7.
    Cohen, D.A.: A New Class of Binary CSPs for which Arc-Constistency Is a Decision Procedure. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 807–811. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Cooper, M.C., Jeavons, P.G., Salamon, A.Z.: Generalizing constraint satisfaction on trees: hybrid tractability and variable elimination. Artificial Intelligence 174(9-10), 570–584 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cooper, M.C., Živný, S.: Hybrid tractability of valued constraint problems. Artificial Intelligence 175(9-10), 1555–1569 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cooper, M.C., Živný, S.: Hierarchically nested convex VCSP. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 187–194. Springer, Heidelberg (2011)Google Scholar
  11. 11.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dechter, R.: Constraint Processing. Morgan Kaufmann, San Francisco (2003)zbMATHGoogle Scholar
  13. 13.
    Dechter, R., Pearl, J.: Network-based Heuristics for Constraint Satisfaction Problems. Artificial Intelligence 34(1), 1–38 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Downey, R., Fellows, M.: Parametrized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  15. 15.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Feder, T., Vardi, M.: The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing 28(1), 57–104 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)zbMATHGoogle Scholar
  18. 18.
    Goodman, A.W.: On Sets of Acquaintances and Strangers at any Party. The American Mathematical Monthly 66(9), 778–783 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. Journal of the ACM 54(1), 1–24 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jeavons, P.: On the Algebraic Structure of Combinatorial Problems. Theoretical Computer Science 200(1-2), 185–204 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kolmogorov, V., Živný, S.: Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms. Tech. rep. (August 2010)Google Scholar
  22. 22.
    Kolmogorov, V., Živný, S.: The complexity of conservative VCSPs (submitted for publication, 2011)Google Scholar
  23. 23.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. Journal of Computer System Sciences 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lovász, L.: Coverings and colorings of hypergraphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 3–12 (1973)Google Scholar
  25. 25.
    Maffray, F., Preissmann, M.: On the NP-completeness of the k-colorability problem for triangle-free graphs. Discrete Mathematics 162(1-3), 313–317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Papadimitriou, C.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  27. 27.
    Rossi, F., van Beek, P., Walsh, T. (eds.): The Handbook of Constraint Programming. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  28. 28.
    Schaefer, T.: The Complexity of Satisfiability Problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)Google Scholar
  29. 29.
    Schiex, T., Fargier, H., Verfaillie, G.: Valued Constraint Satisfaction Problems: Hard and Easy Problems. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence, IJCAI 1995 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Martin C. Cooper
    • 1
  • Stanislav Živný
    • 2
  1. 1.IRITUniversity of Toulouse IIIToulouseFrance
  2. 2.University CollegeUniversity of OxfordUK

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