Advertisement

Combinatorial Proof that Subprojective Constraint Satisfaction Problems are NP-Complete

  • Jaroslav Nešetřil
  • Mark Siggers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

Abstract

We introduce a new general polynomial-time construction- the fibre construction- which reduces any constraint satisfaction problem \({\rm CSP}(\mathcal H)\) to the constraint satisfaction problem \({\rm CSP}(\mathcal{P})\), where \({\mathcal{P}}\) is any subprojective relational structure. As a consequence we get a new proof (not using universal algebra) that \({\rm CSP}(\mathcal{P})\) is NP-complete for any subprojective (and thus also projective) relational structure. This provides a starting point for a new combinatorial approach to the NP-completeness part of the conjectured Dichotomy Classification of CSPs, which was previously obtained by algebraic methods. This approach is flexible enough to yield NP-completeness of coloring problems with large girth and bounded degree restrictions.

Keywords

Relational Structure Constraint Satisfaction Constraint Satisfaction Problem Homomorphic Image Universal Algebra 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bodnarčuk, V.G., Kaluzhnin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras I - II (in Russian), Kibernetika, 3 (1969), 1-10 and 5 (1969), 1-9. English version: Cybernetics, 243-252 and 531-539 (1969)Google Scholar
  2. 2.
    Bulatov, A.: A dichotomy theorem for constraints on a three element set. FOCS 2002, 649–658 (2002)Google Scholar
  3. 3.
    Bulatov, A.: Tractable conservative Constraint Satisfaction Problems, ACM Trans. on Comp. Logic. LICS 2003, 321–330 (2003)Google Scholar
  4. 4.
    Bulatov, A.: H-Coloring Dichotomy Revisited. Theoret. Comp. Sci. 349(1), 31–39 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the complexity of constraints using finite algebras. SIAM J. Comput. 34(3), 720–742 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bulatov, A., Jeavons, P., Krokhin, A.: Constraint satisfaction problems and finite algebras. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 272–282. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  7. 7.
    Creignou, N., Khanna, S., Sudan, M.: Complexity Classifications of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, SIAM  (2001)Google Scholar
  8. 8.
    Feder, T., Hell, P., Huang, J.: List Homomorphisms of Graphs with Bounded Degree (submitted)Google Scholar
  9. 9.
    Feder, T., Vardi, M.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1999)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Geiger, D.: Closed systems of functions and predicates. Pacific. Journal of Math. 27, 95–100 (1968)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hell, P.: Algorithmic aspects of graph homomorphisms. In: Survey in Combinatorics 2003, pp. 239–276. Cambridge University Press, Cambridge (2003)Google Scholar
  12. 12.
    Hell, P.: From Graph Colouring to Constraint Satisfaction: There and Back Again. In: Klazar, M., Kratochvil, J., Loebl, M., Matousek, J., Thomas, R., Valtr, P. (eds.) Topics in Discrete Mathematics. Dedicated to Jarik Nesetril on the Occasion of his 60th Birthday, pp. 407–432. Springer, Heidelberg (2006)Google Scholar
  13. 13.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Combin. Theory B 48, 92–100 (1990)zbMATHCrossRefGoogle Scholar
  14. 14.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  15. 15.
    Jeavons, P.G.: On the algebraic structure of combinatorial problems. Theoret. Comput. Sci. 200(1-2), 185–204 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Jeavons, P.G., Cohen, D.A., Gyssens, M.: Closure properties of Constraints. Journal of the ACM 44, 527–548 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kostochka, A., Nešetřil, J., Smolíková, P.: Colorings and homomorphisms of degenerate and bounded degree graphs. Graph theory (Prague, 1998). Discrete Math 233(1-3), 257–276 (2001)zbMATHGoogle Scholar
  18. 18.
    Larose, B., Zádori, L.: The Complexity of the Extendibility Problem for Finite Posets. SIAM J. Discrete Math. 17(1), 114–121 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Łuczak, T., Nešetřil, J.: A probabilistic approach to the dichotomy problem. SIAM J. Comput. 36(3), 835–843 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Montanari, U.: Networks of constraints: Fundamental properties and applications to picture processing. Information Sciences 7, 95–132 (1974)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Müller, V.: On colorings of graphs without short cycles. Discrete Math. 26, 165–176 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Nešetřil, J., Rödl, V.: Chromatically optimal rigid graphs. J. Comb. Th. B 46, 122–141 (1989)Google Scholar
  23. 23.
    Nešetřil, J., Siggers, M.: A New Combinatorial Approach to the CSP Dichotomy Classification (submitted, 2007)Google Scholar
  24. 24.
    Nešetřil, J., Zhu, X.: On sparse graphs with given colorings and homomorphisms. J. Combin. Theory Ser. B 90(1), 161–172 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pippenger, N.: Theories of Computability. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  26. 26.
    Pöschel, R., Kalužnin, L.A.: Funktionen- und Relatrionenalgebren. DVW, Berlin (1979)Google Scholar
  27. 27.
    McKenzie, R.: Personal CommunicationGoogle Scholar
  28. 28.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the 10th ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226 (1978)Google Scholar
  29. 29.
    Siggers, M.: On Highly Ramsey Infinte Graphs. (submitted, 2006)Google Scholar
  30. 30.
    Siggers, M.: Dichotomy for Bounded Degree H-colouring (submitted, 2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jaroslav Nešetřil
    • 1
  • Mark Siggers
    • 1
  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI), Charles University Malostranské nám. 25, 11800 Praha 1Czech Republic

Personalised recommendations