A Polynomial Kernel for Trivially Perfect Editing

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)

Abstract

We give a kernel with O(k7) vertices for Trivially Perfect Editing, the problem of adding or removing at most k edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao (Social Networks, 35(3):439–450, 2013) and by Liu, Wang, and Guo (Tsinghua Science and Technology, 19(4):346–357, 2014). Using our technique one can also obtain kernels of the same size for the related problems, Trivially Perfect Completion and Trivially Perfect Deletion.

We complement our study of Trivially Perfect Editing by proving that, contrary to Trivially Perfect Completion, it cannot be solved in time 2o(k)·nO(1) unless the Exponential Time Hypothesis fails. In this manner we complete the picture of the parameterized and kernelization complexity of the classic edge modification problems for the class of trivially perfect graphs.

Keywords

Polynomial Kernel Reduction Rule Perfect Graph Edge Deletion Induce Subgraph 
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.
    Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. J. Comput. Syst. Sci. 76(7), 524–531 (2010)CrossRefMATHGoogle Scholar
  2. 2.
    Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Applied Mathematics 154(13), 1824–1844 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters 58(4), 171–176 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cai, L., Cai, Y.: Incompressibility of H-free edge modification. Algorithmica 71(3), 731–757 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cygan, M., Pilipczuk, M., Pilipczuk, M., van Leeuwen, E.J., Wrochna, M.: Polynomial kernelization for removing induced claws and diamonds. In: WG (2015)Google Scholar
  6. 6.
    Drange, P.G., Dregi, M.S., Lokshtanov, D., Sullivan, B.D.: On the Intractability of Threshold Editing. In: ESA (2015)Google Scholar
  7. 7.
    Drange, P.G., Fomin, F.V., Pilipczuk, M., Villanger, Y.: Exploring subexponential parameterized complexity of completion problems. In: STACS (2014)Google Scholar
  8. 8.
    Drange, P.G., Pilipczuk, M.: A Polynomial Kernel for Trivially Perfect Editing. CoRR, abs/1412.7558 (2014)Google Scholar
  9. 9.
    Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar F-deletion: Approximation, kernelization and optimal FPT algorithms. In: FOCS (2012)Google Scholar
  10. 10.
    Guillemot, S., Havet, F., Paul, C., Perez, A.: On the (non-)existence of polynomial kernels for P -free edge modification problems. Algorithmica 65(4), 900–926 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Guo, J.: Problem kernels for NP-complete edge deletion problems: Split and related graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 915–926. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63(4), 512–530 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Applied Mathematics 160(15), 2259–2270 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kratsch, S., Wahlström, M.: Two edge modification problems without polynomial kernels. Discrete Optimization 10(3), 193–199 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, Y., Wang, J., Guo, J.: An overview of kernelization algorithms for graph modification problems. Tsinghua Science and Technology 19(4), 346–357 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nastos, J., Gao, Y.: Familial groups in social networks. Social Networks 35(3), 439–450 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Dept. InformaticsUniv. BergenBergenNorway
  2. 2.Inst. InformaticsUniv. WarsawWarsawPoland

Personalised recommendations