New Plain-Exponential Time Classes for Graph Homomorphism

  • Magnus Wahlström
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5675)


A homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f: V(G) →V(H) such that if uv ∈ E(G) then f(u)f(v) ∈ E(H). The problem Hom (G,H) of deciding whether there is a homomorphism is NP-complete, and in fact the fastest known algorithm for the general case has a running time of O*n(H) cn(G), for a constant 0 < c < 1. In this paper, we consider restrictions on the graphs G and H such that the problem can be solved in plain-exponential time, i.e. in time O*c n(G) + n(H) for some constant c. Previous research has identified two such restrictions. If H = K k or contains K k as a core (i.e. a homomorphically equivalent subgraph), then Hom (G,H) is the k-coloring problem, which can be solved in time O*2 n(G) (Björklund, Husfeldt, Koivisto); and if H has treewidth at most k, then Hom (G,H) can be solved in time O*(k + 3) n(G) (Fomin, Heggernes, Kratsch, 2007). We extend these results to cases of bounded cliquewidth: if H has cliquewidth at most k, then we can count the number of homomorphisms from G to H in time O*(2k + 1) max (n(G),n(H)), including the time for finding a k-expression for H. The result extends to deciding HomG,H) when H has a core with a k-expression, in this case with a somewhat worse running time.

If G has cliquewidth at most k, then a similar result holds, with a worse dependency on k: We are able to count Hom (G,H) in time roughly O*(2k + 1) n(G) + 22kn(H), and this also extends to when G has a core of cliquewidth at most k with a similar running time.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Björklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion–exclusion. SIAM J. Comput. (to appear); prelim. versions in FOCS 2006Google Scholar
  2. 2.
    Byskov, J.M.: Exact Algorithms for Graph Colouring and Exact Satisfiability. PhD thesis, University of Aarhus (2005)Google Scholar
  3. 3.
    Corneil, D.G., Habib, M., Lanlignel, J.-M., Reed, B.A., Rotics, U.: Polynomial time recognition of clique-width ≤ 3 graphs (extended abstract). In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 126–134. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Courcelle, B., Engelfriet, J., Rozenberg, G.: Handle-rewriting hypergraph grammars. J. Comput. System Sci. 46, 218–270 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Applied Mathematics 101(1-3), 77–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.: Parameterized Complexity. Monographs in Computer Science. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  7. 7.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. J. Graph Algorithms Appl. 7(2), 131–140 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width minimization is NP-hard. In: STOC 2006, pp. 354–362 (2006)Google Scholar
  9. 9.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  10. 10.
    Fomin, F.V., Heggernes, P., Kratsch, D.: Exact algorithms for graph homomorphisms. Theory of Comput. Syst. 41(2), 381–393 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)Google Scholar
  12. 12.
    Gutin, G., Hell, P., Rafiey, A., Yeo, A.: A dichotomy for minimum cost graph homomorphisms. European J. Combin. 29, 900–911 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gutin, G., Rafiey, A., Yeo, A.: Minimum cost and list homomorphisms to semicomplete digraphs. Discrete Applied Mathematics 154(6), 890–897 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gutin, G., Rafiey, A., Yeo, A., Tso, M.: Level of repair analysis and minimum cost homomorphisms of graphs. Discrete Applied Mathematics 154(6), 881–889 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hell, P., Nešetřil, J.: On the complexity of H-coloring. J. Comb. Theory, Ser. B 48(1), 92–110 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)CrossRefMATHGoogle Scholar
  17. 17.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jonsson, P., Nordh, G., Thapper, J.: The maximum solution problem on graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 228–239. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Lawler, E.: A note on the complexity of the chromatic number problem. Information Processing Letters 5, 66–67 (1976)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Traxler, P.: The time complexity of constraint satisfaction. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 190–201. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Williams, R.: A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci. 348(2-3), 357–365 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Magnus Wahlström
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

Personalised recommendations