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Subset Feedback Vertex Set Is Fixed-Parameter Tractable

  • Marek Cygan
  • Marcin Pilipczuk
  • Michał Pilipczuk
  • Jakub Onufry Wojtaszczyk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

The classical Feedback Vertex Set problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. Feedback Vertex Set has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field.

In this paper we consider a more general and difficult version of the problem, named Subset Feedback Vertex Set (Subset-FVS in short) where an instance comes additionally with a set S ⊆ V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis Subset-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP’00, SIDMA’00].

The question whether the Subset-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2 k n O(1) instances with the size of S bounded by O(k 3), using kernelization techniques such as the 2-Expansion Lemma, Menger’s theorem and Gallai’s theorem. These two facts allow us to give a 2 O(klogk) n O(1) time algorithm solving the Subset Feedback Vertex Set problem, proving that it is indeed fixed-parameter tractable.

Keywords

Feasible Solution Undirected Graph Simple Cycle Quadratic Kernel Edge Bubble 
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.

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References

  1. 1.
    Becker, A., Bar-Yehuda, R., Geiger, D.: Randomized algorithms for the loop cutset problem. J. Artif. Intell. Res (JAIR) 12, 219–234 (2000)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., van Dijk, T.C.: A cubic kernel for feedback vertex set and loop cutset. Theory Comput. Syst. 46(3), 566–597 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bousquet, N., Daligault, J., Thomassé, S.: Multicut is FPT. In: Proc. of STOC 2011 (to appear, 2011)Google Scholar
  5. 5.
    Burrage, K., Estivill-Castro, V., Fellows, M.R., Langston, M.A., Mac, S., Rosamond, F.A.: The undirected feedback vertex set problem has a poly(k) kernel. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 192–202. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    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
  8. 8.
    Chen, J., Liu, Y., Lu, S., O’Sullivan, B., Razgon, I.: A fixed-parameter algorithm for the directed feedback vertex set problem. In: Proc. of STOC 2008, pp. 177–186 (2008)Google Scholar
  9. 9.
    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. CoRR abs/1103.0534 (2011)Google Scholar
  10. 10.
    Dehne, F.K.H.A., Fellows, M.R., Langston, M.A., Rosamond, F.A., Stevens, K.: An O(2\(^{\mbox{{o}(k)}}\)n\(^{\mbox{3}}\)) FPT algorithm for the undirected feedback vertex set problem. In: Wang, L. (ed.) COCOON 2005. LNCS, vol. 3595, pp. 859–869. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Demaine, E.D., Hajiaghayi, M.T., Marx, D.: Open problems from dagstuhl seminar 09511 (2009)Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Fixed parameter tractability and completeness. In: Complexity Theory: Current Research, pp. 191–225 (1992)Google Scholar
  13. 13.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  14. 14.
    Even, G., Naor, J., Schieber, B., Sudan, M.: Approximating minimum feedback sets and multicuts in directed graphs. Algorithmica 20(2), 151–174 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Even, G., Naor, J., Schieber, B., Zosin, L.: Approximating minimum subset feedback sets in undirected graphs with applications. SIAM J. Discrete Math. 13(2), 255–267 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Even, G., Naor, J., Zosin, L.: An 8-approximation algorithm for the subset feedback vertex set problem. SIAM J. Comput. 30(4), 1231–1252 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Guillemot, S.: FPT algorithms for path-transversals and cycle-transversals problems in graphs. In: Grohe, M., Niedermeier, R. (eds.) IWPEC 2008. LNCS, vol. 5018, pp. 129–140. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Guo, J., Gramm, J., Hüffner, F., Niedermeier, R., Wernicke, S.: Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J. Comput. Syst. Sci. 72(8), 1386–1396 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kanj, I.A., Pelsmajer, M.J., Schaefer, M.: Parameterized algorithms for feedback vertex set. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 235–247. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Kawarabayashi, K., Kobayashi, Y.: Fixed-parameter tractability for the subset feedback set problem and the S-cycle packing problem (2010) (manuscript)Google Scholar
  21. 21.
    Marx, D.: Parameterized graph separation problems. Theor. Comput. Sci. 351(3), 394–406 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Marx, D., Razgon, I.: Fixed-parameter tractability of multicut parameterized by the size of the cutset. In: Proc. of STOC 2011 (to appear, 2011)Google Scholar
  23. 23.
    Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for undirected feedback vertex set. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 241–248. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  24. 24.
    Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transactions on Algorithms 2(3), 403–415 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Reed, B.A., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Thomassé, S.: A quadratic kernel for feedback vertex set. In: Proc. of SODA 2009, pp. 115–119 (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Marek Cygan
    • 1
  • Marcin Pilipczuk
    • 1
  • Michał Pilipczuk
    • 1
  • Jakub Onufry Wojtaszczyk
    • 2
  1. 1.Institute of InformaticsUniversity of WarsawPoland
  2. 2.Institute of MathematicsUniversity of WarsawPoland

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