Skip to main content
Log in

List H-Coloring a Graph by Removing Few Vertices

  • Published:
Algorithmica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The notation \(x_1\rightarrow x_2\rightarrow \dots \rightarrow x_\ell \) is a shorthand for \((x_1 \rightarrow x_2) \wedge (x_2 \rightarrow x_3) \wedge \cdots \wedge (x_{\ell -1} \rightarrow x_\ell )\).

  2. For background on MSO, we refer the reader to, e.g., the textbook of Flum and Grohe [15]; very briefly, MSO is a logical language that allows quantification over the elements and subsets of the universe of a relational structure.

References

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: On multiway cut parameterized above lower bounds. ACM Trans. Comput. Theory 5(1), 3 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, New York (2013)

    Book  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feder, T., Hell, P.: List homomorphisms to reflexive graphs. J. Comb. Theory Ser. B 72(2), 236–250 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feder, T., Hell, P., Huang, J.: List homomorphisms and circular arc graphs. Combinatorica 19(4), 487–505 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feder, T., Hell, P., Huang, J.: Bi-arc graphs and the complexity of list homomorphisms. J. Graph Theory 42(1), 61–80 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Feder, T., Hell, P., Huang, J.: List homomorphisms of graphs with bounded degrees. Discrete Math. 307, 386–392 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM 49(6), 716–752 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006)

    MATH  Google Scholar 

  16. Gutin, G., Rafiey, A., Yeo, A.: Minimum cost and list homomorphisms to semicomplete digraphs. Discrete Appl. Math. 154, 890–897 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  18. Hell, P., Nešetřil, J.: On the complexity of \(H\)-coloring. J. Comb. Theory Ser. B 48, 92–110 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lokshtanov, D., Marx, D.: Clustering with local restrictions. Inf. Comput. 222, 278–292 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  24. Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Marx, D.: Parameterized complexity and approximation algorithms. Comput. J. 51(1), 60–78 (2008)

    Article  Google Scholar 

  26. Marx, D., O’Sullivan, B., Razgon, I.: Finding small separators in linear time via treewidth reduction. ACM Trans. Algorithms 9(4), 30 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  29. Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. J. Comput. Syst. Sci. 75(8), 435–450 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Spinrad, J.: Circular-arc graphs with clique cover number two. J. Comb. Theory Ser. B 44(3), 300–306 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for their comments that helped improve the presentation, and for pointing out a problem that required some technical work to fix.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dániel Marx.

Additional information

Supported by ERC Starting Grant PARAMTIGHT (No. 280152) and OTKA Grant NK105645.

R. Chitnis: Most of this work was done when the author was a student at University of Maryland, College Park. Supported in part by NSF CAREER award 1053605, NSF Grant CCF-1161626, ONR YIP award N000141110662, DARPA/AFOSR Grant FA9550-12-1-0423, a University of Maryland Research and Scholarship Award (RASA) and a Summer International Research Fellowship from University of Maryland.

L. Egri: Supported by NSERC and FQRNT.

A preliminary version of the paper appeared in the proceedings of ESA 2013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chitnis, R., Egri, L. & Marx, D. List H-Coloring a Graph by Removing Few Vertices. Algorithmica 78, 110–146 (2017). https://doi.org/10.1007/s00453-016-0139-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0139-6

Keywords

Navigation