Linear Kernels and Single-Exponential Algorithms via Protrusion Decompositions

  • Eun Jung Kim
  • Alexander Langer
  • Christophe Paul
  • Felix Reidl
  • Peter Rossmanith
  • Ignasi Sau
  • Somnath Sikdar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

Abstract

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs.

Let \(\mathcal{F}\) be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar\(\mathcal{F}\)- Deletion asks whether G has a set X ⊆ V(G) such that \(|X|\leqslant k\) and G − X is H-minor-free for every \(H\in \mathcal{F}\). As our second application, we present the first single-exponential algorithm to solve Planar\(\mathcal{F}\)- Deletion. Namely, our algorithm runs in time 2O(k)·n2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family \(\mathcal{F}\).

Keywords

parameterized complexity linear kernels algorithmic meta-theorems sparse graphs single-exponential algorithms graph minors 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for Dominating Set. Journal of the ACM 51, 363–384 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: Dynamic programming on graphs with bounded treewidth. In: Lepistö, T., Salomaa, A. (eds.) ICALP 1988. LNCS, vol. 317, pp. 105–118. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (Meta) Kernelization. In: Proc. of 50th FOCS, pp. 629–638. IEEE Computer Society (2009)Google Scholar
  5. 5.
    Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Information and Computation 167(2), 86–119 (2001)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. Journal of Computer and System Sciences 74(7), 1188–1198 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., Wojtaszczyk, J.O.: An improved FPT algorithm and quadratic kernel for pathwidth one vertex deletion. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 95–106. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Dehne, F., Fellows, M., Langston, M.A., Rosamond, F., Stevens, K.: An O(2O(k) n 3) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 859–869. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory, 4th edn. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Dinneen, M.: Too many minor order obstructions. Journal of Universal Computer Science 3(11), 1199–1206 (1997)MathSciNetMATHGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)Google Scholar
  12. 12.
    Fellows, M.R., Langston, M.A.: Nonconstructive tools for proving polynomial-time decidability. Journal of the ACM 35, 727–739 (1988)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and kernelization. In: STACS 28th. LIPIcs, vol. 9, pp. 189–200. Schloss Dagstuhl–Leibniz-Zentrum fu (2011)̈r Informatik (2011)Google Scholar
  14. 14.
    Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar \(\mathcal{F}\)-Deletion: Approximation and Optimal FPT Algorithms. In: Proc. of 53rd FOCS, pp. 470–479. IEEE Computer Society (2012)Google Scholar
  15. 15.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proc. of 21st SODA, pp. 503–510. SIAM (2010)Google Scholar
  16. 16.
    Fomin, F.V., Oum, S., Thilikos, D.M.: Rank-width and tree-width of H-minor-free graphs. European Journal of Combinatorics 31(7), 1617–1628 (2010)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    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)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Guo, J., Niedermeier, R.: Linear problem kernels for NP-hard problems on planar graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Joret, G., Paul, C., Sau, I., Saurabh, S., Thomassé, S.: Hitting and harvesting pumpkins. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 394–407. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Kim, E.J., Paul, C., Philip, G.: A single-exponential FPT-algorithm for K 4-minor cover problem. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 119–130. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Lokshtanov, D., Saurabh, S., Sikdar, S.: Simpler parameterized algorithm for OCT. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 380–384. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. 22.
    Philip, G., Raman, V., Villanger, Y.: A quartic kernel for Pathwidth-One Vertex Deletion. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 196–207. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Robertson, N., Seymour, P.D.: Graph minors II. Algorithmic aspects of tree-width. Journal of Algorithms 7, 309–322 (1986)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Robertson, N., Seymour, P.D.: Graph minors XIII. The disjoint paths problem. Journal of Combinatorial Theory, Series B 63, 65–110 (1995)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Thomason, A.: The extremal function for complete minors. Journal of Combinatorial Theory, Series B 81(2), 318–338 (2001)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eun Jung Kim
    • 1
  • Alexander Langer
    • 2
  • Christophe Paul
    • 3
  • Felix Reidl
    • 2
  • Peter Rossmanith
    • 2
  • Ignasi Sau
    • 3
  • Somnath Sikdar
    • 2
  1. 1.CNRS, LAMSADEParisFrance
  2. 2.Theoretical Computer Science, Department of Computer ScienceRWTH Aachen UniversityGermany
  3. 3.CNRS, LIRMMMontpellierFrance

Personalised recommendations