Constraint Satisfaction Parameterized by Solution Size

  • Andrei A. Bulatov
  • Dániel Marx
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

In the constraint satisfaction problem (CSP) corresponding to a constraint language (i.e., a set of relations) Γ, the goal is to find an assignment of values to variables so that a given set of constraints specified by relations from Γ is satisfied. In this paper we study the fixed-parameter tractability of constraint satisfaction problems parameterized by the size of the solution in the following sense: one of the possible values, say 0, is “free,” and the number of variables allowed to take other, “expensive,” values is restricted. A size constraint requires that exactly k variables take nonzero values. We also study a more refined version of this restriction: a global cardinality constraint prescribes how many variables have to be assigned each particular value. We study the parameterized complexity of these types of CSPs where the parameter is the required number k of nonzero variables. As special cases, we can obtain natural and well-studied parameterized problems such as Independent set, Vertex Cover, d -Hitting Set, Biclique, etc. In the case of constraint languages closed under substitution of constants, we give a complete characterization of the fixed-parameter tractable cases of CSPs with size constraints, and we show that all the remaining problems are W[1]-hard. For CSPs with cardinality constraints, we obtain a similar classification, but for some of the problems we are only able to show that they are Biclique-hard. The exact parameterized complexity of the Biclique problem is a notorious open problem, although it is believed to be W[1]-hard.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bessière, C., Hebrard, E., Hnich, B., Walsh, T.: The complexity of global constraints. In: Wallace, M. (ed.) AAAI. LNCS, vol. 3258, pp. 112–117. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Bulatov, A.: Tractable conservative constraint satisfaction problems. In: LICS, pp. 321–330. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  3. 3.
    Bulatov, A.A., Jeavons, P., Krokhin, A.A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34(3), 720–742 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bulatov, A.A., Marx, D.: The complexity of global cardinality constraints. In: LICS, pp. 419–428. IEEE Computer Society, Los Alamitos (2009)Google Scholar
  5. 5.
    Creignou, N., Schnoor, H., Schnoor, I.: Non-uniform boolean constraint satisfaction problems with cardinality constraint. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 109–123. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity (1999)Google Scholar
  7. 7.
    Feder, T., Vardi, M.Y.: Monotone monadic snp and constraint satisfaction. In: STOC, pp. 612–622 (1993)Google Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)MATHGoogle Scholar
  9. 9.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. J. ACM 44, 527–548 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Jeavons, P., Cohen, D., Gyssens, M.: How to determine the expressive power of constraints. Constraints 4, 113–131 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM J. Comput. 30(6), 1863–1920 (2001)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kratsch, S., Wahlström, M.: Preprocessing of min ones problems: A dichotomy. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 653–665. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Krokhin, A.A., Marx, D.: On the hardness of losing weight. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 662–673. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Marx, D.: Parameterized complexity of constraint satisfaction problems. Computational Complexity 14Google Scholar
  15. 15.
    Régin, J.C., Gomes, C.P.: The cardinality matrix constraint. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 572–587. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Rosenberg, I.: Multiple-valued hyperstructures. In: ISMVL, pp. 326–333 (1998)Google Scholar
  17. 17.
    Schaefer, T.J.: The complexity of satisfiability problems. In: STOC, pp. 216–226 (1978)Google Scholar
  18. 18.
    Szeider, S.: The parameterized complexity of k-flip local search for sat and max sat. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 276–283. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Andrei A. Bulatov
    • 1
  • Dániel Marx
    • 2
  1. 1.Simon Fraser UniversityCanada
  2. 2.Humboldt-Universität zu BerlinGermany

Personalised recommendations