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Undirected Feedback Vertex Set

  • Reference work entry
  • First Online:
Encyclopedia of Algorithms

  • 396 Accesses

Years and Authors of Summarized Original Work

  • 2005; Dehne, Fellows, Langston, Rosamond, Stevens Guo, Gramm, Hüffner, Niedermeier, Wernicke

Problem Definition

The Undirected Feedback Vertex Set (UFVS) problem is defined as follows:

Input: :

An undirected graph \( { G = (V, E) } \) and an integer \( { k\ge 0 } \).

Task: :

Find a feedback vertex set\( { F\subseteq V } \) with \( { |F| \leq k } \) such that each cycle in G contains at least one vertex from F. (The removal of all vertices in F from G results in a forest.)

Karp [11] showed that UFVS is NP-complete. Lund and Yannakakis [12] proved that there exists some constant \( { \epsilon > 0 } \) such that it is NP-hard to approximate the optimization version of UFVS to within a factor of \( { 1\,+\,\epsilon } \). The best-known polynomial-time approximation algorithm for UFVS has a factor of 2 [1, 4]. There is a simple and elegant randomized algorithm due to Becker et al. [3] which solves UFVS in O(c·4k·kn) time on an n-vertex and m-edg...

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Recommended Reading

  1. Bafna V, Berman P, Fujito T (1999) A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J Discret Math 3(2):289–297

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  2. Bar-Yehuda R, Geiger D, Naor J, Roth RM (1998) Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM J Comput 27(4):942–959

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  3. Becker A, Bar-Yehuda R, Geiger D (2000) Randomized algorithms for the loop cutset problem. J Artif Intell Res 12:219–234

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  4. Becker A, Geiger D (1994) Approximation algorithms for the Loop Cutset problem. In: Proceedings of the 10th conference on uncertainty in artificial intelligence. Morgan Kaufman, San Fransisco, pp 60–68

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  5. Bodlaender HL (1994) On disjoint cycles. Int J Found Comput Sci 5(1):59–68

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  6. Dehne F, Fellows MR, Langston MA, Rosamond F Stevens K (2005) An O(2O(k)n3) FPT algorithm for the undirected feedback vertex set problem. In: Proceedings of the 11th COCOON. LNCS, vol 3595. Springer, Berlin, pp 859–869. Long version to appear in: J Discret Algorithm

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  7. Downey RG, Fellows MR (1992) Fixed-parameter tractability and completeness. Congr Numerant 87:161–187

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  8. Downey RG, Fellows MR (1999) Parameterized complexity. Springer, Heidelberg

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  9. Fomin FV, Gaspers S, Pyatkin AV (2006) Finding a minimum feedback vertex set in time O(1.7548n). In: Proceedings of the 2th IWPEC. LNCS, vol 4196. Springer, Berlin, pp 184–191

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  10. Guo J, Gramm J, Hüffner F, Niedermeier R, Wernicke S (2006) Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization. J Comput Syst Sci 72(8):1386–1396

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  11. Karp R (1972) Reducibility among combinatorial problems. In: Miller R, Thatcher J (eds) Complexity of computer computations. Plenum Press, New York, pp 85–103

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  12. Lund C, Yannakakis M (1993) The approximation of maximum subgraph problems. In: Proceedings of the 20th ICALP. LNCS, vol 700. Springer, Berlin, pp 40–51

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  14. Reed B, Smith K, Vetta A (2004) Finding odd cycle transversals. Oper Res Lett 32(4):299–301

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Acknowledgments

Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Jena, Jena, Germany

    Jiong Guo

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  1. Jiong Guo
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Correspondence to Jiong Guo .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Guo, J. (2016). Undirected Feedback Vertex Set. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_450

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