Advertisement

A Quartic Kernel for Pathwidth-One Vertex Deletion

  • Geevarghese Philip
  • Venkatesh Raman
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

The pathwidth of a graph is a measure of how path-like the graph is. Given a graph G and an integer k, the problem of finding whether there exist at most k vertices in G whose deletion results in a graph of pathwidth at most one is NP-Complete. We initiate the study of the parameterized complexity of this problem, parameterized by k. We show that the problem has a quartic vertex-kernel: We show that, given an input instance (G = (V,E),k);|V| = n, we can construct, in polynomial time, an instance (G′,k′) such that (i) (G,k) is a YES instance if and only if (G′,k′) is a YES instance, (ii) G′ has \({\mathcal O}(k^{4})\) vertices, and (iii) k′ ≤ k. We also give a fixed parameter tractable (FPT) algorithm for the problem that runs in \({\mathcal O}(7^{k}k\cdot n^{2})\) time.

Keywords

Parameterized Complexity Polynomial Kernel Tree Decomposition Layout Problem Reduction Rule 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Álvarez, C., Serna, M.: The Proper Interval Colored Graph problem for caterpillar trees. Electronic Notes in Discrete Mathematics 17, 23–28 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnborg, S., Proskurowski, A., Seese, D.: Monadic Second Order Logic, Tree Automata and Forbidden Minors. In: Schönfeld, W., Börger, E., Kleine Büning, H., Richter, M.M. (eds.) CSL 1990. LNCS, vol. 533, pp. 1–16. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  3. 3.
    Assmann, S.F., Peck, G.W., Sysło, M.M., Zak, J.: The bandwidth of caterpillars with hairs of length 1 and 2. SIAM Journal on Algebraic and Discrete Methods 2(4), 387–393 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bafna, V., Berman, P., Fujito, T.: A 2-Approximation Algorithm for the Undirected Feedback Vertex Set problem. SIAM Journal of Discrete Mathematics 12(3), 289–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L.: On disjoint cycles. International Journal of Foundations of Computer Science 5(1), 59–68 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science 209(1–2), 1–45 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bodlaender, H.L.: Treewidth: Characterizations, Applications, and Computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Bodlaender, H.L.: A Cubic Kernel for Feedback Vertex Set. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 320–331. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. 9.
    Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial Optimization on Graphs of Bounded Treewidth. The Computer Journal 51(3), 255–269 (2008)CrossRefGoogle Scholar
  10. 10.
    Bodlaender, H.L., van Dijk, T.C.: A cubic kernel for feedback vertex set and loop cutset. Theory of Computing Systems 46(3), 566–597 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bryant, R.L., Kinnersley, N.G., Fellows, M.R., Langston, M.A.: On Finding Obstruction Sets and Polynomial-Time Algorithms for Gate Matrix Layout. In: Proceedings of the 25th Allerton Conference on Communication, Control and Computing, pp. 397–398 (1987)Google Scholar
  12. 12.
    Burrage, K., Estivill Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The Undirected Feedback Vertex Set Problem Has a Poly(k) Kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 192–202. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Cao, Y., Chen, J., Liu, Y.: On Feedback Vertex Set: New Measure and New Structures. In: Kaplan, H. (ed.) Algorithm Theory - SWAT 2010. LNCS, vol. 6139, pp. 93–104. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: An improved fpt algorithm and quadratic kernel for pathwidth one vertex deletion. Accepted at IPEC 2010 (2010)Google Scholar
  15. 15.
    Dehne, F., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O(2O(k) n 3) FPT-Algorithm for the Undirected Feedback Vertex Set problem. Theory of Computing Systems 41(3), 479–492 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  17. 17.
    Dorn, F., Fomin, F.V., Thilikos, D.M.: Subexponential parameterized algorithms. Computer Science Review 2(1), 29–39 (2008)CrossRefzbMATHGoogle Scholar
  18. 18.
    Downey, R.G., Fellows, M.R.: Fixed Parameter Tractability and Completeness. In: Complexity Theory: Current Research, pp. 191–225. Cambridge University Press, Cambridge (1992)Google Scholar
  19. 19.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  20. 20.
    Fellows, M.R., Langston, M.A.: On Search, Decision and the Efficiency of Polynomial-time Algorithms. In: Proceedings of STOC 1989, pp. 501–512. ACM Press, New York (1989)Google Scholar
  21. 21.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  22. 22.
    Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. Journal of Computer and System Sciences 72(8), 1386–1396 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  24. 24.
    Kanj, I.A., Pelsmajer, M.J., Schaefer, M.: Parameterized Algorithms for Feedback Vertex Set. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 235–247. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  25. 25.
    Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Communications, 85–103 (1972)Google Scholar
  26. 26.
    Kinnersley, N.G., Langston, M.A.: Obstruction Set Isolation for the Gate Matrix Layout problem. Discrete Applied Mathematics 54(2-3), 169–213 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kloks, T.: Treewidth – computations and approximations. LNCS, vol. 842. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  28. 28.
    Lin, M., Lin, Z., Xu, J.: Graph bandwidth of weighted caterpillars. Theoretical Computer Science 363(3), 266–277 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. Journal of Computer and System Sciences 20(2), 219–230 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  31. 31.
    Papadimitriou, C.H.: The NP-Completeness of the bandwidth minimization problem. Computing 16(3), 263–270 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Philip, G., Raman, V., Villanger, Y.: A quartic kernel for pathwidth-one vertex deletion. A full version of the current paper, http://www.imsc.res.in/~gphilip/publications/pwone.pdf
  33. 33.
    Raman, V., Saurabh, S., Subramanian, C.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transactions on Algorithms 2(3), 403–415 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Robertson, N., Seymour, P.D.: Graph minors I. Excluding a forest. Journal of Combinatorial Theory, Series B 35(1), 39–61 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Robertson, N., Seymour, P.D.: Graph Minors. II. Algorithmic Aspects of Tree-Width. Journal of Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proceedings of SODA 2009, Society for Industrial and Applied Mathematics, pp. 115–119 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Geevarghese Philip
    • 1
  • Venkatesh Raman
    • 1
  • Yngve Villanger
    • 2
  1. 1.The Institute of Mathematical SciencesChennaiIndia
  2. 2.University of BergenBergenNorway

Personalised recommendations