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Kernelization Hardness of Connectivity Problems in d-Degenerate Graphs

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

A graph is d-degenerate if its every subgraph contains a vertex of degree at most d. For instance, planar graphs are 5-degenerate. Inspired by recent work by Philip, Raman and Sikdar, who have shown the existence of a polynomial kernel for Dominating Set in d-degenerate graphs, we investigate kernelization hardness of problems that include connectivity requirement in this class of graphs.

Our main contribution is the proof that Connected Dominating Set does not admit a polynomial kernel in d-degenerate graphs for d ≥ 2 unless the polynomial hierarchy collapses up to the third level. We prove this using a problem originated from bioinformatics – Colourful Graph Motif – analyzed and proved to be NP-hard by Fellows et al. This problem nicely encapsulates the hardness of the connectivity requirement in kernelization. Our technique yields also an alternative proof that, under the same complexity assumption, Steiner Tree does not admit a polynomial kernel. The original proof, via reduction from Set Cover, is due to Dom, Lokshtanov and Saurabh.

We extend our analysis by showing that, unless \(PH = \Sigma_p^3\), there do not exist polynomial kernels for Steiner Tree, Connected Feedback Vertex Set and Connected Odd Cycle Transversal in d-degenerate graphs for d ≥ 2. On the other hand, we show a polynomial kernel for Connected Vertex Cover in graphs that do not contain the biclique K i,j as a subgraph.

Keywords

Planar Graph STEINER Tree Vertex Cover Polynomial Kernel Free Graph 
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.

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References

  1. 1.
    Alon, N., Gutner, S.: Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 394–405. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels (extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M. (Meta) kernelization. In: Proc. of FOCS 2009, pp. 629–638 (2009)Google Scholar
  4. 4.
    Bodlaender, H.L., Thomasse, S., Yeo, A.: Analysis of data reduction: Transformations give evidence for non-existence of polynomial kernels, technical Report UU-CS-2008-030, Institute of Information and Computing Sciences, Utrecht University, Netherlands (2008)Google Scholar
  5. 5.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: Further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through colors and IDs. In: Proc. of ICALP 2009, pp. 378–389 (2009)Google Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999), http://citeseer.ist.psu.edu/downey98parameterized.html CrossRefzbMATHGoogle Scholar
  8. 8.
    Escoffier, B., Gourvès, L., Monnot, J.: Complexity and approximation results for the connected vertex cover problem. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 202–213. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Fellows, M.R., Fertin, G., Hermelin, D., Vialette, S.: Sharp tractability borderlines for finding connected motifs in vertex-colored graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 340–351. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Fernau, H., Fomin, F.V., Lokshtanov, D., Raible, D., Saurabh, S., Villanger, Y.: Kernel(s) for problems with no kernel: On out-trees with many leaves. In: Proc. of STACS 2009, pp. 421–432 (2009)Google Scholar
  11. 11.
    Fomin, F., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proc. of SODA 2010, pp. 503–510 (2010)Google Scholar
  12. 12.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: Proc. of STOC 2008, pp. 133–142 (2008)Google Scholar
  13. 13.
    Golovach, P.A., Villanger, Y.: Parameterized complexity for domination problems on degenerate graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 195–205. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Kostochka, A.V.: Lower bound of the hadwiger number of graphs by their average degree. Combinatorica 4(4), 307–316 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Misra, N., Philip, G., Raman, V., Saurabh, S., Sikdar, S.: FPT Algorithms for Connected Feedback Vertex Set. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 269–280. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Philip, G., Raman, V., Sikdar, S.: Solving dominating set in larger classes of graphs: Fpt algorithms and polynomial kernels. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 694–705. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Thomason, A.: An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc. 95(2), 261–265 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Thomason, A.: The extremal function for complete minors. J. Comb. Theory, Ser. B 81(2), 318–338 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proc. of SODA 2009, pp. 115–119 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 1
  • Michał Pilipczuk
    • 1
  • Jakub Onufry Wojtaszczyk
    • 1
  1. 1.Faculty of Mathematics, Computer Science and MechanicsUniversity of WarsawPoland

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