NP by Means of Lifts and Shadows

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


We show that every NP problem is polynomially equivalent to a simple combinatorial problem: the membership problem for a special class of digraphs. These classes are defined by means of shadows (projections) and by finitely many forbidden colored (lifted) subgraphs. Our characterization is motivated by the analysis of syntactical subclasses with the full computational power of NP, which were first studied by Feder and Vardi.

Our approach applies to many combinatorial problems and it induces the characterization of coloring problems (CSP) defined by means of shadows. This turns out to be related to homomorphism dualities. We prove that a class of digraphs (relational structures) defined by finitely many forbidden colored subgraphs (i.e. lifted substructures) is a CSP class if and only if all the the forbidden structures are homomorphically equivalent to trees. We show a surprising richness of coloring problems when restricted to most frequent graph classes. Using results of Nešetřil and Ossona de Mendez for bounded expansion classes (which include bounded degree and proper minor closed classes) we prove that the restriction of every class defined as the shadow of finitely many colored subgraphs equals to the restriction of a coloring (CSP) class.


Digraph homomorphism duality NP Constraint Satisfaction Problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Atserias, A.: On Digraph Coloring Problems and Treewidth Duality. In: 20th IEEE Symposium on Logic in Computer Science (LICS), pp. 106–115 (2005)Google Scholar
  2. 2.
    Atserias, A., Dawar, A., Kolaitis, P.G.: On Preservation under Homomorphisms and Conjunctive Queries. Journal of the ACM 53(2), 208–237 (2006)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Erdős, P., Füredi, Z., Hajnal, A., Komjáth, P., Rödl, V., Seress, Á.: Coloring graphs with locally few colors. Discrete Math 59, 21–34 (1986)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets. In: Karp, R. (ed.) Complexity of Computation, SIAM-AMS Proceedings, vol. 7, pp. 43–73 (1974)Google Scholar
  5. 5.
    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
  6. 6.
    Foniok, J., Nešetřil, J., Tardif, C.: Generalized dualities and maximal finite antichains in the homomorphism order of relational structures. In: KAM-DIMATIA Series 2006-766 (to appear in European J. Comb.)Google Scholar
  7. 7.
    Gács, P., Lovász, L.: Some remarks on generalized spectra. Z. Math. Log. Grdl. 23(6), 547–554 (1977)zbMATHCrossRefGoogle Scholar
  8. 8.
    Feder, T., Hell, P., Klein, S., Motwani, R.: Complexity of graph partition problems. In: 31st Annual ACM STOC, pp. 464–472 (1999)Google Scholar
  9. 9.
    Hell, P., Nešetřil, J.: Graphs and Homomorphism. Oxford University Press, Oxford (2004)Google Scholar
  10. 10.
    Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16, 760–778 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kun, G.: On the complexity of Constraint Satisfaction Problem, PhD thesis (in Hungarian) (2006)Google Scholar
  12. 12.
    Kun, G.: Constraints, MMSNP and expander structures, Combinatorica (submitted, 2007)Google Scholar
  13. 13.
    Kun, G., Nešetřil, J.: Forbidden lifts (NP and CSP for combinatorists). In: KAM-DIMATIA Series 2006-775 (to appear in European J. Comb.)Google Scholar
  14. 14.
    Ladner, R.E.: On the structure of Polynomial Time Reducibility. Journal of the ACM 22(1), 155–171 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Luczak, T., Nešetřil, J.: A probabilistic approach to the dichotomy problem. SIAM J. Comp. 36(3), 835–843 (2006)zbMATHCrossRefGoogle Scholar
  16. 16.
    Madelaine, F.: Constraint satisfaction problems and related logic, PhD thesis (2003)Google Scholar
  17. 17.
    Madelaine, F., Stewart, I.A.: Constraint satisfaction problems and related logic, (manuscript, 2005)Google Scholar
  18. 18.
    Matoušek, J., Nešetřil, J.: Constructions of sparse graphs with given homomorphisms (to appear)Google Scholar
  19. 19.
    Nešetřil, J., Pultr, A.: On classes of relations and graphs determined by subobjects and factorobjects. Discrete Math. 22, 287–300 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Nešetřil, J., de Mendez, P.O.: Low tree-width decompositions and algorithmic consequences. In: STOC 2006, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 391–400. ACM Press, New York (2006)CrossRefGoogle Scholar
  21. 21.
    Nešetřil, J., de Mendez, P.O.: Grad and Classes with bounded expansion III. - Restricted Dualities, KAM-DIMATIA Series 2005-741 (to appear in European J. Comb.)Google Scholar
  22. 22.
    Nešetřil, J., Rödl, V.: Chromatically optimal rigid graphs. J. Comb. Th. B 46, 133–141 (1989)zbMATHCrossRefGoogle Scholar
  23. 23.
    Nešetřil, J., Tardif, C.: Duality theorems for finite structures (characterising gaps and good characterization. J. Combin. Theory B 80, 80–97 (2000)zbMATHCrossRefGoogle Scholar
  24. 24.
    Nešetřil, J., Zhu, X.: On sparse graphs with given colorings and homomorphisms. J. Comb. Th. B 90(1), 161–172 (2004)zbMATHCrossRefGoogle Scholar
  25. 25.
    Rossman, B.: Existential positive types and preservation under homomorphisms. In: 20th IEEE Symposium on Logic in Computer Science (LICS 2005), pp. 467–476 (2005)Google Scholar
  26. 26.
    Simonyi, G., Tardos, G.: Local chromatic number, Ky Fan’s theorem and circular colorings. Combinatorica 26, 589–626 (2006)MathSciNetGoogle Scholar
  27. 27.
    Vardi, M.Y.: The complexity of relational query languages. In: Proceedings of 14th ACM Symposium on Theory of Computing  pp. 137–146 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gábor Kun
    • 1
  • Jaroslav Nešetřil
    • 1
  1. 1.Department of Mathematics, University of Memphis, 373 Dunn Hall, Memphis, TN 38152, Department of Applied Mathematics (KAM) and, Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám 22, Praha 

Personalised recommendations