WALCOM 2010: WALCOM: Algorithms and Computation pp 269-280

# FPT Algorithms for Connected Feedback Vertex Set

• Neeldhara Misra
• Geevarghese Philip
• Venkatesh Raman
• Saket Saurabh
• Somnath Sikdar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5942)

## Abstract

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G = (V,E) and an integer k, decide whether there exists F ⊆ V, |F| ≤ k, such that G[V ∖ F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 O(k) n O(1)) on general graphs and in time $$O(2^{O(\sqrt{k}\log k)}n^{O(1)})$$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.

## Keywords

Vertex Cover Input Graph Compact Representation Polynomial Kernel Tree Decomposition
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.

## References

1. 1.
Bang-Jensen, J., Gutin, G.Z.: Digraphs: Theory, Algorithms and Applications, 2nd edn. Springer, Heidelberg (2009)
2. 2.
Bodlaender, H.L.: On disjoint cycles. In: Schmidt, G., Berghammer, R. (eds.) WG 1991. LNCS, vol. 570, pp. 230–238. Springer, Heidelberg (1992)Google Scholar
3. 3.
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25(6), 1305–1317 (1996)
4. 4.
Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels (extended abstract). In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 563–574. Springer, Heidelberg (2008)
5. 5.
Chen, J., Fomin, F.V., Liu, Y., Lu, S., Villanger, Y.: Improved algorithms for the feedback vertex set problems. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 422–433. Springer, Heidelberg (2007)
6. 6.
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)
7. 7.
Demaine, E.D., Hajiaghayi, M.: Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica 28(1), 19–36 (2008)
8. 8.
Demaine, E.D., Hajiaghayi, M., ichi Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation, and coloring. In: Proceedings of FOCS 2005, pp. 637–646. IEEE Computer Society, Los Alamitos (2005)Google Scholar
9. 9.
Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005)
10. 10.
Ding, B., Yu, J.X., Wang, S., Qin, L., Zhang, X., Lin, X.: Finding top-k min-cost connected trees in databases. In: ICDE, pp. 836–845. IEEE, Los Alamitos (2007)Google Scholar
11. 11.
Dom, M., Lokshtanov, D., Saurabh, S.: Incompressibility through Colors and IDs. In: Albers, S., et al. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 378–389. Springer, Heidelberg (2009)Google Scholar
12. 12.
Festa, P., Pardalos, P.M., Resende, M.G.: Feedback set problems. In: Handbook of Combinatorial Optimization, pp. 209–258. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
13. 13.
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
14. 14.
Fomin, F.V., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. Algorithmica 52(2), 153–166 (2008)
15. 15.
Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: Proceedings of SODA 2010 (2010) (to appear)Google Scholar
16. 16.
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)
17. 17.
Mölle, D., Richter, S., Rossmanith, P.: Enumerate and expand: Improved algorithms for connected vertex cover and tree cover. Theory of Computing Systems 43(2), 234–253 (2008)
18. 18.
Moser, H.: Exact algorithms for generalizations of vertex cover. Master’s thesis, Institut für Informatik, Friedrich-Schiller-Universität (2005)Google Scholar
19. 19.
Nederlof, J.: Fast polynomial-space algorithms using möbius inversion: Improving on steiner tree and related problems. In: Albers, S., et al. (eds.) ICALP 2009, pp. 713–725. Springer, Heidelberg (2009)Google Scholar
20. 20.
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)
21. 21.
Sitters, R., Grigoriev, A.: Connected feedback vertex set in planar graphs. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911. Springer, Heidelberg (2009)Google Scholar
22. 22.
Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proceedings of SODA 2009, pp. 115–119. Society for Industrial and Applied Mathematics (2009)Google Scholar

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• Neeldhara Misra
• 1
• Geevarghese Philip
• 1
• Venkatesh Raman
• 1
• Saket Saurabh
• 1
• Somnath Sikdar
• 1
1. 1.The Institute of Mathematical SciencesIndia