Skip to main content
SpringerLink
Log in

On Feedback Vertex Set: New Measure and New Structures

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We present a new parameterized algorithm for the feedback vertex set problem (fvs) on undirected graphs. We approach the problem by considering a variation of it, the disjoint feedback vertex set problem (disjoint-fvs), which finds a feedback vertex set of size \(k\) that has no overlap with a given feedback vertex set \(F\) of the graph \(G\). We develop an improved kernelization algorithm for disjoint-fvs and show that disjoint-fvs can be solved in polynomial time when all vertices in \(G{\setminus }F\) have degrees upper bounded by three. We then propose a new branch-and-search process on disjoint-fvs, and introduce a new branch-and-search measure. The process effectively reduces a given graph to a graph on which disjoint-fvs becomes polynomial-time solvable, and the new measure more accurately evaluates the efficiency of the process. These algorithmic and combinatorial studies enable us to develop an \(O^*(3.83^k)\)-time parameterized algorithm for the general fvs problem, improving all previous algorithms for the problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. Following the recent convention in the research in exact and parameterized algorithms, we will denote by \(O^*(f(k))\) the complexity \(O(f(k)n^{O(1)})\) for a super-polynomial function \(f\).

References

  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)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  3. Becker, A., Geiger, D.: Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83(1), 167–188 (1996)

    Article  MathSciNet  Google Scholar 

  4. Bodlaender, H.L.: On disjoint cycles. Int. J. Found. Comput. Sci. 5(1), 59–68 (1994)

    Article  MATH  Google Scholar 

  5. Chen, J.: Minimum and maximum imbeddings. In: Gross, J.L., Yellen, J. (eds.) Handbook of Graph Theory, pp. 625–641. CRC Press, Boca Raton (2003)

    Google Scholar 

  6. Chen, J., Fomin, F.V., Liu, Y., Villanger, Y.: Improved algorithms for feedback vertex set problems. J. Comput. System Sci. 74(7), 1188–1198 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cygan, M., Nederlof, J., Pilipczuk, M., Pilipczuk, M.ł, van Rooij, J.M.M., Wojtaszczyk, O.J.: Solving connectivity problems parameterized by treewidth in single exponential time. In Rafail Ostrovsky, editor, FOCS, pages 150–159. IEEE, 2011. Full version is available as arXiv:1103.0534

  8. Dehne, F.K.H.A., Fellows, M.R., Langston, Michael A., Rosamond, F.A., Stevens, K.: An O(\(2^{O(k)} n^3\)) FPT algorithm for the undirected feedback vertex set problem. Theory Comput. Systems 41(3), 479–492 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Bidmensional structures: algorithms, combinatorics and logic (Dagstuhl Seminar 13121). Dagstuhl Rep. 3(3), 51–74 (2013)

    Google Scholar 

  10. Downey, R.G., Fellows, M.R.: Fixed parameter tractability and completeness. Complexity Theory: Current Research, pp. 191–225. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  11. Downey, R.G., Fellows, M.R.:. Fundamentals of Parameterized Complexity. Undegraduate texts in computer science. Springer (2013)

  12. Erdős, P., Pósa, L.: On the maximal number of disjoint circuits of a graph. Publ. Math. Debr. 9, 3–12 (1962)

    Google Scholar 

  13. Fellows, M.R., Langston, M.A.: Nonconstructive tools for proving polynomial-time decidability. J. ACM 35(3), 727–739 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fomin, F.V., Gaspers, S., Pyatkin, A.V., Razgon, I.: On the minimum feedback vertex set problem: Exact and enumeration algorithms. Algorithmica 52(2), 293–307 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Furst, M.L., Gross, J.L., McGeoch, L.A.: Finding a maximum-genus graph imbedding. J. ACM 35(3), 523–534 (1988)

    Article  MathSciNet  Google Scholar 

  16. Gabow, H.N., Xu, Y.: Efficient theoretic and practical algorithms for linear matroid intersection problems. Journal of Computer and System Sciences, 53(1):129–147 (1996). A preliminary version appeared in FOCS 1989

  17. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  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. System Sci. 72, 1386–1396 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kanj, I.A., Pelsmajer, M.J., Schaefer, M.: Parameterized algorithms for feedback vertex set. In: Rodney G. Downey, Michael R. Fellows, and Frank K.H.A. Dehne (eds), IWPEC, volume 3162 of LNCS. pp. 235–247. Springer, (2004)

  20. Kociumaka, T., Pilipczuk, M.: Faster deterministic feedback vertex set. CoRR arXiv:1306.3566[cs.DS], (2013)

  21. Lokshtanov, D., Marx, Dá, Saurabh, S.: Lower bounds based on the Exponential Time Hypothesis. Bulletin EATCS 105, 41–72 (2011)

    MATH  MathSciNet  Google Scholar 

  22. Lovász, László: The matroid matching problem. In Algebraic Methods in Graph Theory, volume 25 of Colloquia Mathematica Societatis János Bolyai, pp. 495–517. Szeged (1980)

  23. Raman, V., Saurabh, S., Subramanian, C.R.: Faster fixed parameter tractable algorithms for undirected feedback vertex set. In Prosenjit Bose and Pat Morin, editors, ISAAC, volume 2518 of LNCS, p 241–248. Springer, (2002)

  24. Raman, V., Saurabh, S.: Faster fixed parameter tractable algorithms for finding feedback vertex sets. ACM Transac. Algorithm. 2(3), 403–415 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Speckenmeyer, E. Untersuchungen zum Feedback Vertex Set Problem in ungerichteten Graphen. PhD thesis, Universität-GH Paderborn, Reihe Informatik, Bericht, (1983)

  27. Speckenmeyer, E.: On feedback vertex sets and nonseparating independent sets in cubic graphs. J. Graph Theory 12(3), 405–412 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  28. Ueno, S., Kajitani, Y., Gotoh, S.: On the nonseparating independent set problem and feedback set problem for graphs with no vertex degree exceeding three. Discret. Math. 72(1–3), 355–360 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

All authors were supported in part by the US National Science Foundation under grants CCF-0830455 and CCF-0917288. The first author was supported in part by the European Research Council (ERC) grant 280152 and the Hungarian Scientific Research Fund (OTKA) grant NK105645. We would like to thank anonymous referees for thoughtful and detailed comments, which led to an improved presentation.

Author information

Authors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, TX , 77843-3112, USA

    Jianer Chen

  3. Supercomputing Facility, Texas A&M University, College Station, TX , 77845, USA

    Yang Liu

Authors
  1. Yixin Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianer Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yixin Cao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, Y., Chen, J. & Liu, Y. On Feedback Vertex Set: New Measure and New Structures. Algorithmica 73, 63–86 (2015). https://doi.org/10.1007/s00453-014-9904-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-014-9904-6

Keywords

Search

Navigation