On Parameterized Independent Feedback Vertex Set

  • Neeldhara Misra
  • Geevarghese Philip
  • Venkatesh Raman
  • Saket Saurabh
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

Abstract

We investigate a generalization of the classical Feedback Vertex Set (FVS) problem from the point of view of parameterized algorithms. Independent Feedback Vertex Set (IFVS) is the “independent” variant of the FVS problem and is defined as follows: given a graph G and an integer k, decide whether there exists F ⊆ V(G), |F| ≤ k, such that G[V(G) ∖ F] is a forest and G[F] is an independent set; the parameter is k. Note that the similarly parameterized versions of the FVS problem — where there is no restriction on the graph G[F] — and its connected variant CFVS — where G[F] is required to be connected — have been extensively studied in the literature. The FVS problem easily reduces to the IFVS problem in a manner that preserves the solution size, and so any algorithmic result for IFVS directly carries over to FVSA. We show that IFVS can be solved in time O(5knO(1)) time where n is the number of vertices in the input graph G, and obtain a cubic (O(k3)) kernel for the problem. Note the contrast with the CFVS problem, which does not admit a polynomial kernel unless CoNP ⊆ NP/Poly.

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References

  1. 1.
    Alon, N., Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: Solving max-r-sat above a tight lower bound. In: SODA, pp. 511–517 (2010)Google Scholar
  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)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: Kernelization: New upper and lower bound techniques. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 17–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: Meta kernelization. In: FOCS, pp. 629–638 (2009)Google Scholar
  5. 5.
    Cao, Y., Chen, J., Liu, Y.: On feedback vertex set new measure and new structures. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 93–104. Springer, Heidelberg (2010)CrossRefGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, J., Liu, Y., Lu, S., O’sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. Journal of the ACM 55(5), 21:1–21:19 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Fixed parameter tractability and completeness. In: Complexity Theory: Current Research, pp. 191–225. Cambridge University Press, Cambridge (1992)Google Scholar
  10. 10.
    Fellows, M.R., Guo, J., Moser, H., Niedermeier, R.: A generalization of nemhauser and trotter’s local optimization theorem. In: STACS, pp. 409–420 (2009)Google Scholar
  11. 11.
    Festa, P., Pardalos, P.M., Resende, M.G.: Feedback set problems. In: Handbook of Combinatorial Optimization, pp. 209–258. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  12. 12.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. In: Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)Google Scholar
  13. 13.
    Fomin, F.V., Lokshtanov, D., Misra, N., Philip, G., Saurabh, S.: Hitting forbidden minors: Approximation and Kernelization. In: Proc. of the 28th Symposium on Theoretical Aspects of Computer Science, STACS (to appear, 2011), http://arxiv.org/abs/1010.1365
  14. 14.
    Fomin, F.V., Grandoni, F., Kratsch, D.: Solving connected dominating set faster than 2n. Algorithmica 52(2), 153–166 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fomin, F.V., Lokshtanov, D., Saurabh, S., Thilikos, D.M.: Bidimensionality and kernels. In: SODA, pp. 503–510 (2010)Google Scholar
  16. 16.
    Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  18. 18.
    Gutin, G., Kim, E.J., Szeider, S., Yeo, A.: A probabilistic approach to problems parameterized above or below tight bounds. J. Comput. Syst. Sci. 77(2), 422–429 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kratsch, S.: Polynomial kernelizations for MIN F\(^{\mbox{+}}\) Pi\(_{\mbox{1}}\) and MAX NP. In: STACS, pp. 601–612 (2009)Google Scholar
  20. 20.
    Lokshtanov, D., Mnich, M., Saurabh, S.: Linear kernel for planar connected dominating set. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 281–290. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Marx, D., O’Sullivan, B., Razgon, I.: Treewidth reduction for constrained separation and bipartization problems. In: STACS, pp. 561–572 (2010)Google Scholar
  22. 22.
    Misra, N., Philip, G., Raman, V., Saurabh, S., Sikdar, S.: FPT algorithms for connected feedback vertex set. In: Rahman, M. S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 269–280. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  23. 23.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, vol. 31. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  24. 24.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 115–119. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefGoogle Scholar
  26. 26.
    Thomassé, S.: A 4k 2 kernel for feedback vertex set. ACM Transactions on Algorithms 6, 32:1–32:8 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Neeldhara Misra
    • 1
  • Geevarghese Philip
    • 1
  • Venkatesh Raman
    • 1
  • Saket Saurabh
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia

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