On Cutwidth Parameterized by Vertex Cover

  • Marek Cygan
  • Daniel Lokshtanov
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)

Abstract

We study the Cutwidth problem, where input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 k n O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 n/2 n O(1)) time algorithm for Cutwidth on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for Cutwidth on a graph class where the problem remains NP-complete. Additionally, we show that Cutwidth parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless \(\ensuremath{\textrm{NP} \subseteq \textrm{coNP}/\textrm{poly}}\). Our kernelization lower bound contrasts the recent result of Bodlaender et al.[ICALP 2011] that Treewidth parameterized by vertex cover does admit a polynomial kernel.

Keywords

Bipartite Graph Vertex Cover Hamiltonian Path Input Graph Polynomial Kernel 
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.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adolphson, D., Hu, T.C.: Optimal linear ordering. SIAM J. Appl. Math. 25, 403–423 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellman, R.: Dynamic programming treatment of the travelling salesman problem. J. ACM 9(1), 61–63 (1962)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Björklund, A.: Determinant sums for undirected hamiltonicity. In: FOCS, pp. 173–182 (2010)Google Scholar
  4. 4.
    Blin, G., Fertin, G., Hermelin, D., Vialette, S.: Fixed-parameter algorithms for protein similarity search under mrna structure constraints. Journal of Discrete Algorithms 6, 618–626 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Cross-composition: A new technique for kernelization lower bounds. In: Schwentick, T., Dürr, C. (eds.) STACS. LIPIcs, vol. 9, pp. 165–176. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2011)Google Scholar
  6. 6.
    Bodlaender, H.L., Jansen, B.M.P., Kratsch, S.: Preprocessing for Treewidth: A Combinatorial Analysis through Kernelization. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 437–448. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Botafogo, R.A.: Cluster analysis for hypertext systems. In: SIGIR, pp. 116–125 (1993)Google Scholar
  9. 9.
    Chung, M., Makedon, F., Sudborough, I., Turner, J.: Polynomial time algorithms for the min cut problem on degree restricted trees. SIAM Journal on Computing 14, 158–177 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Diaz, J., Penrose, M., Petit, J., Serna, M.: Approximating layout problems on random geometric graphs. Journal of Algorithms 39, 78–117 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph Layout Problems Parameterized by Vertex Cover. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 294–305. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Fomin, F.V., Kratsch, D., Todinca, I., Villanger, Y.: Exact algorithms for treewidth and minimum fill-in. SIAM J. Comput. 38(3), 1058–1079 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct PCPs for NP. In: STOC 2008: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 133–142. ACM (2008)Google Scholar
  14. 14.
    Gavril, F.: Some np-complete problems on graphs, pp. 91–95 (1977)Google Scholar
  15. 15.
    Heggernes, P., van ’t Hof, P., Lokshtanov, D., Nederlof, J.: Computing the Cutwidth of Bipartite Permutation Graphs in Linear Time. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 75–87. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Heggernes, P., Lokshtanov, D., Mihai, R., Papadopoulos, C.: Cutwidth of Split Graphs, Threshold Graphs, and Proper Interval Graphs. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 218–229. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10(1), 196–210 (1962)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jansen, B.M.P., Kratsch, S.: Data reduction for graph coloring problems. CoRR abs/1104.4229 (2011)Google Scholar
  19. 19.
    Junguer, M., Reinelt, G., Rinaldi, G.: The travelling salesman problem. In: Handbook on Operations Research and Management Sciences, pp. 225–330 (1995)Google Scholar
  20. 20.
    Karger, D.R.: A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM J. Comput. 29(2), 492–514 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Karp, R.M.: Dynamic programming meets the principle of inclusion and exclusion. Oper. Res. Lett. 1, 49–51 (1982)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Leighton, F., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM 46, 787–832 (1999)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Makedon, F., Sudborough, I.H.: On minimizing width in linear layouts. Discrete Applied Mathematics 23, 243–265 (1989)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Monien, B., Sudborough, I.H.: Min cut is np-complete for edge weighted trees. Theoretical Computer Science 58, 209–229 (1988)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mutzel, P.: A Polyhedral Approach to Planar Augmentation and Related Problems. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 494–507. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  26. 26.
    Suchan, K., Villanger, Y.: Computing Pathwidth Faster than 2. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 324–335. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  27. 27.
    Thilikos, D.M., Serna, M.J., Bodlaender, H.L.: Cutwidth ii: Algorithms for partial w-trees of bounded degree. Journal of Algorithms 56, 24–49 (2005)MathSciNetMATHGoogle Scholar
  28. 28.
    Yannakakis, M.: A polynomial algorithm for the min cut linear arrangement of trees. Journal of the ACM 32, 950–988 (1985)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Yuan, J., Zhou, S.: Optimal labelling of unit interval graphs. Appl. Math. J. Chinese Univ. Ser. B (English edition) 10, 337–344 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Marek Cygan
    • 1
  • Daniel Lokshtanov
    • 2
  • Marcin Pilipczuk
    • 1
  • Michał Pilipczuk
    • 1
  • Saket Saurabh
    • 3
  1. 1.Institute of InformaticsUniversity of WarsawPoland
  2. 2.University of CaliforniaSan Diego, La JollaUSA
  3. 3.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations