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FPT Algorithms for FVS Parameterized by Split and Cluster Vertex Deletion Sets and Other Parameters

  • Diptapriyo MajumdarEmail author
  • Venkatesh Raman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10336)

Abstract

A feedback vertex set in an undirected graph is a subset of vertices whose deletion results in an acyclic graph. The problem (which we call FVS) of finding a minimum (or k sized) feedback vertex set is NP-hard in general graphs, while it is polynomial time solvable in some classes of graphs including split graphs and cluster graphs. The current best fixed-parameter tractable (FPT) algorithm for determining whether a given undirected graph has a feedback vertex set of size at most k has a runtime of \({\mathcal O}^*(3.618^k)\)(\({\mathcal O}^*\) notation hides polynomial factors). We consider the parameterized complexity of feedback vertex set parameterized by (vertex deletion) distance to some polynomially solvable classes of graphs including cluster and split graphs. We call a graph G a (ci)-graph if its vertex set can be partitioned into c cliques and i independent sets. When \(c=0\) and \(i=2\), such a graph is simply a bipartite graph where FVS is NP-hard. It can be deduced easily that FVS is NP-hard even for constant c when \(i \ge 2\). When \(c \le 1\) and \(i \le 1\), then the graph is a split graph where FVS is solvable in polynomial time. Given a graph, let k be the size of the modulator whose deletion results in a (ci)-graph. We address the parameterized complexity of FVS parameterized by k when \(i \le 1\). Specifically we show that
  1. 1.

    FVS admits an FPT algorithm that runs in \({\mathcal O}^*(3.148^k)\) time, when \(c \le 1\) and \(i \le 1\) (i.e. when the modulator is a deletion set to a split graph). When \(c \ge 2\), we generalize the algorithm to one with runtime \({\mathcal O}(3.148^{k+c}\cdot n^{{\mathcal O}(c)})\). We also show that FVS is W[1]-hard when parameterized by c (i.e. the c in the exponent of n is unavoidable) if \(i \le 1\) extending a known hardness reduction for the case when \(i=0\).

     
  2. 2.

    For the special case when \(i=0\) and \(c \ge 1\), and when there are no edges across vertices in different parts (i.e. the modulator is a deletion set to a cluster graph), we give an \({\mathcal O}^*(5^k)\) algorithm.

     

Keywords

Polynomial Time Vertex Cover Reduction Rule Cluster Graph Split Graph 
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.

References

  1. 1.
    Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math. 12(3), 289–297 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L., Cygan, M., Kratsch, S., Nederlof, J.: Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth. Inf. Comput. 243, 86–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boral, A., Cygan, M., Kociumaka, T., Pilipczuk, M.: A fast branching algorithm for cluster vertex deletion. Theory Comput. Syst. 58(2), 357–376 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cao, Y., Chen, J., Liu, Y.: On feedback vertex set: new measure and new structures. Algorithmica 73(1), 63–86 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. Syst. Sci. 74(7), 1188–1198 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chen, J., Kanj, I.A., Jia, W.: Vertex cover: further observations and further improvements. J. Algorithms 41(2), 280–301 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cygan, M., Pilipczuk, M.: Split Vertex Deletion meets Vertex Cover: new fixed-parameter and exact exponential-time algorithms. Inf. Process. Lett. 113(5–6), 179–182 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  10. 10.
    Jansen, B., Raman, V., Vatshelle, M.: Parameter ecology for feedback vertex set. Tsinghua Sci. Technol. 19(4), 387–409 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jansen, B.M.P., Fellows, M.R., Rosamond, F.A.: Towards fully multivariate algorithmics: parameter ecology and the deconstruction of computational complexity. Eur. J. Comb. 34(3), 541–566 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. Inf. Process. Lett. 114(10), 556–560 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kolay, S., Panolan, F.: Parameterized algorithms for deletion to (r, ell)-graphs. In: Proceedings of FSTTCS, pp. 420–433 (2015)Google Scholar
  14. 14.
    Kolay, S., Panolan, F.: Parameterized algorithms for deletion to (r, l)-graphs. CoRR, abs/1504.08120 (2015)Google Scholar
  15. 15.
    Lokshtanov, D., Narayanaswamy, N.S., Raman, V., Ramanujan, M.S., Saurabh, S.: Faster parameterized algorithms using linear programming. ACM Trans. Algorithms 11(2), 15:1–15:31 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Majumdar, D.: Structural parameterizations of feedback vertex set. In: IPEC, Aarhus, Denmark, pp. 21:1–21:16 (2016)Google Scholar
  17. 17.
    Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed-parameter tractable algorithms for finding feedback vertex sets. ACM Trans. Algorithms 2(3), 403–415 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Thomassé, S.: A 4\(k^{2}\) kernel for feedback vertex set. ACM Trans. Algorithms 6(2), 1–8 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.The Institute of Mathematical Sciences, HBNIChennaiIndia

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