Advertisement

Theory of Computing Systems

, Volume 63, Issue 3, pp 587–614 | Cite as

On the Parameterized Complexity of Contraction to Generalization of Trees

  • Akanksha AgarwalEmail author
  • Saket Saurabh
  • Prafullkumar Tale
Article
  • 39 Downloads

Abstract

For a family of graphs \(\mathcal {F}\), the \(\mathcal {F}\)-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists SE(G) of size at most k such that G/S belongs to \(\mathcal {F}\). Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al. [Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied \(\mathcal {F}\)-Contraction when \(\mathcal {F}\) is a simple family of graphs such as trees and paths. In this paper, we study the \(\mathcal {F}\)-Contraction problem, where \(\mathcal {F}\) generalizes the family of trees. In particular, we define this generalization in a “parameterized way”. Let \(\mathbb {T}_{\ell }\) be the family of graphs such that each graph in \(\mathbb {T}_{\ell }\) can be made into a tree by deleting at most edges. Thus, the problem we study is \(\mathbb {T}_{\ell }\)-Contraction. We design an FPT algorithm for \(\mathbb {T}_{\ell }\)-Contraction running in time \(\mathcal {O}((2\sqrt {\ell }+ 2)^{\mathcal {O}(k + \ell )} \cdot n^{\mathcal {O}(1)})\). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for \(\mathbb {T}_{\ell }\)-Contraction of size \( \mathcal {O}([k(k + 2\ell )]^{(\lceil {\frac {\alpha }{\alpha -1}\rceil + 1)}})\).

Keywords

Graph contraction Fixed parameter tractability Graph algorithms Generalization of trees 

Notes

Acknowledgments

A preliminary version of this manuscript has been accepted at the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017).

The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreements no. 306992 (PARAPPROX).

References

  1. 1.
    Agrawal, A., Kanesh, L., Saurabh, S., Tale, P.: Paths to Trees and Cacti. In: CIAC, pp. 31–42 (2017)Google Scholar
  2. 2.
    Agrawal, A., Lokshtanov, D., Saurabh, S., Zehavi, M.: Split Contraction: The Untold Story. In: STACS, LIPIcs, vol. 66, pp. 5:1–5:14 (2017)Google Scholar
  3. 3.
    Asano, T., Hirata, T.: Edge-Contraction problems. J. Comput. Syst. Sci. 26(2), 197–208 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Belmonte, R., Golovach, P.A., Hof, P., Paulusma, D.: Parameterized complexity of three edge contraction problems with degree constraints. Acta Informatica 51(7), 473–497 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci. 75(8), 423–434 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. Theor. Comput. Sci. 412(35), 4570–4578 (2011).  https://doi.org/10.1016/j.tcs.2011.04.039 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cai, L., Guo, C.: Contracting Few Edges to Remove Forbidden Induced Subgraphs. In: IPEC, pp. 97–109 (2013)Google Scholar
  9. 9.
    Cygan, M.: Deterministic Parameterized Connected Vertex Cover. In: Scandinavian Workshop on Algorithm Theory, pp. 95–106. Springer (2012)Google Scholar
  10. 10.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms. Springer, Berlin (2015)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M., van Rooij, J.M.M., Wojtaszczyk, J.O.: Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time. In: IEEE 52Nd Annual Symposium on Foundations of Computer Science, FOCS, pp. 150–159 (2011)Google Scholar
  12. 12.
    Diestel, R.: Graph Theory, 4th Edition, Graduate texts in mathematics, vol. 173. Springer, Berlin (2012)Google Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Berlin (1997)zbMATHGoogle Scholar
  14. 14.
    Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized complexity. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized complexity theory. Texts in theoretical computer science. an EATCS series. Springer, Berlin (2006)Google Scholar
  16. 16.
    Fortnow, L., Santhanam, R.: Infeasibility of instance compression and succinct pcps for NP. J. Comput. Syst. Sci. 77(1), 91–106 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Golovach, P.A., van ’t Hof, P., Paulusma, D.: Obtaining planarity by contracting few edges. Theor. Comput. Sci. 476, 38–46 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guillemot, S., Marx, D.: A faster FPT algorithm for bipartite contraction. Inf. Process. Lett. 113(22–24), 906–912 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Heggernes, P., van ’t Hof, P., Lokshtanov, D., Paul, C.: Obtaining a bipartite graph by contracting few edges. SIAM J. Discret. Math. 27(4), 2143–2156 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Heggernes, P., van ’t Hof, P., Lévêque, B., Lokshtanov, D., Paul, C.: Contracting graphs to paths and trees. Algorithmica 68(1), 109–132 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, W., Feng, Q., Chen, J., Hu, S.: Improved kernel results for some FPT problems based on simple observations. Theor. Comput. Sci. 657, 20–27 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lokshtanov, D., Misra, N., Saurabh, S.: On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges. In: IPEC, pp. 243–254 (2013)Google Scholar
  23. 23.
    Lokshtanov, D., Panolan, F., Ramanujan, M.S., Saurabh, S.: Lossy kernelization. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, pp. 224–237 (2017)Google Scholar
  24. 24.
    Naor, M., Schulman, L.J., Srinivasan, A.: Splitters and Near-Optimal Derandomization. In: 1995. Proceedings., 36Th Annual Symposium On Foundations of Computer Science, pp. 182–191. IEEE (1995)Google Scholar
  25. 25.
    Niedermeier, R.: Invitation to fixed-parameter algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)CrossRefGoogle Scholar
  26. 26.
    Watanabe, T., Ae, T., Nakamura, A.: On the removal of forbidden graphs by edge-deletion or by edge-contraction. Discret. Appl. Math. 3(2), 151–153 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Watanabe, T., Ae, T., Nakamura, A.: On the NP-hardness of edge-deletion and-contraction problems. Discret. Appl. Math. 6(1), 63–78 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.The Institute of Mathematical Sciences, HBNIChennaiIndia

Personalised recommendations