Algorithmica

, Volume 78, Issue 1, pp 110–146

List H-Coloring a Graph by Removing Few Vertices

Article
  • 102 Downloads

Abstract

In the deletion version of the list homomorphism problem, we are given graphs G and H, a list \(L(v)\subseteq V(H)\) for each vertex \(v\in V(G)\), and an integer k. The task is to decide whether there exists a set \(W \subseteq V(G)\) of size at most k such that there is a homomorphism from \(G {\setminus } W\) to H respecting the lists. We show that DL-Hom(\({H}\)), parameterized by k and |H|, is fixed-parameter tractable for any \((P_6,C_6)\)-free bipartite graph H; already for this restricted class of graphs, the problem generalizes Vertex Cover, Odd Cycle Transversal, and Vertex Multiway Cut parameterized by the size of the cutset and the number of terminals. We conjecture that DL-Hom(\({H}\)) is fixed-parameter tractable for the class of graphs H for which the list homomorphism problem (without deletions) is polynomial-time solvable; by a result of Feder et al. (Combinatorica 19(4):487–505, 1999), a graph H belongs to this class precisely if it is a bipartite graph whose complement is a circular arc graph. We show that this conjecture is equivalent to the fixed-parameter tractability of a single fairly natural satisfiability problem, Clause Deletion Chain-SAT.

Keywords

Fixed-parameter tractability Graph homomorphism Treewidth reduction Iterative compression Shadow removal 

References

  1. 1.
    Chen, Y., Grohe, M., Grüber, M.: On parameterized approximability. In: Parameterized and Exact Computation, Second International Workshop, IWPEC 2006, Zürich, Switzerland, 13–15 Sept 2006, Proceedings, pp. 109–120 (2006)Google Scholar
  2. 2.
    Chitnis, R.H., Cygan, M., Hajiaghayi, M.T., Marx, D.: Directed subset feedback vertex set is fixed-parameter tractable. ACM Trans. Algorithms 11(4), 28 (2015)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chitnis, R.H., Hajiaghayi, M., Marx, D.: Fixed-parameter tractability of directed multiway cut parameterized by the size of the cutset. SIAM J. Comput. 42(4), 1674–1696 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Courcelle, B.: Graph rewriting: an algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science: Volume B: Formal Models and Semantics, pp. 193–242. Elsevier, Amsterdam (1990)Google Scholar
  5. 5.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory 5(1), 3 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, New York (2013)CrossRefMATHGoogle Scholar
  7. 7.
    Egri, L., Hell, P., Larose, B., Rafiey, A.: Space complexity of list H-colouring: a dichotomy. In: Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, 5–7 Jan 2014, pp. 349–365 (2014)Google Scholar
  8. 8.
    Egri, L., Krokhin, A.A., Larose, B., Tesson, P.: The complexity of the list homomorphism problem for graphs. Theory Comput. Syst. 51(2), 143–178 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Feder, T., Hell, P.: List homomorphisms to reflexive graphs. J. Comb. Theory Ser. B 72(2), 236–250 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Feder, T., Hell, P., Huang, J.: List homomorphisms and circular arc graphs. Combinatorica 19(4), 487–505 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Feder, T., Hell, P., Huang, J.: Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory 42(1), 61–80 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Feder, T., Hell, P., Huang, J.: List homomorphisms of graphs with bounded degrees. Discrete Math. 307, 386–392 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: a study through datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM 49(6), 716–752 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)MATHGoogle Scholar
  16. 16.
    Gutin, G., Rafiey, A., Yeo, A.: Minimum cost and list homomorphisms to semicomplete digraphs. Discrete Appl. Math. 154, 890–897 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Hell, P., Nešetřil, J.: On the complexity of \(H\)-coloring. J. Comb. Theory Ser. B 48, 92–110 (1990)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hell, P., Rafiey, A.: The dichotomy of list homomorphisms for digraphs. In: Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, 23–25 Jan 2011, pp. 1703–1713 (2011)Google Scholar
  20. 20.
    Kratsch, S., Pilipczuk, M., Pilipczuk, M., Wahlström, M.: Fixed-parameter tractability of multicut in directed acyclic graphs. SIAM J. Discrete Math. 29(1), 122–144 (2015)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Lokshtanov, D., Marx, D.: Clustering with local restrictions. Inf. Comput. 222, 278–292 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lokshtanov, D., Ramanujan, M.S.: Parameterized tractability of multiway cut with parity constraints. In: Automata, Languages, and Programming—39th International Colloquium, ICALP 2012, Warwick, UK, 9–13 July 2012, Part I, pp. 750–761 (2012)Google Scholar
  24. 24.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)CrossRefGoogle Scholar
  26. 26.
    Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms 9(4), 30 (2013)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. SIAM J. Comput. 43(2), 355–388 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  29. 29.
    Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Spinrad, J.: Circular-arc graphs with clique cover number two. J. Comb. Theory Ser. B 44(3), 300–306 (1988)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Simon Fraser UniversityBurnabyCanada
  3. 3.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

Personalised recommendations