Skip to main content
SpringerLink
Log in

Algorithm for Online 3-Path Vertex Cover

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

A vertex set C of a graph G = (V, E) is a 3-path vertex cover if every path on 3 vertices has at least one vertex in C. This paper studies the online version of the minimum 3-path vertex cover problem, in which vertices are revealed one by one, and one has to determine whether the newly revealed vertex should be chosen into the solution without knowing future information. We show that a natural algorithm has competitive ratio at most Δ, where Δ is the maximum degree of the graph. An example is given showing that the ratio is tight.

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
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Albers, S.: Online algorithms: a survey. Math. Program. 97(1–2), 3–26 (2003)

    Article  MathSciNet  Google Scholar 

  2. Birmelé, E., Delbot, F., Laforest, C.: Mean analysis of an online algorithm for the vertex cover problem. Inf. Process. Lett. 109(9), 436–439 (2009)

    Article  MathSciNet  Google Scholar 

  3. Boyar, J., Favrholdt, L.M., Kudahl, C., Larsen, K.S., Mikkelsen, J.W.: Online algorithms with advice: a survey. ACM Comput. Surv. 47(3), 93–129 (2016)

    MathSciNet  MATH  Google Scholar 

  4. Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum k-path vertex cover. Discret. Appl. Math. 159(12), 1189–1195 (2011)

    Article  MathSciNet  Google Scholar 

  5. Brešar, B., Krivoš-Belluš, R., Semanišin, G., Šparl, P.: On the weighted k-path vertex cover problem. Comput. Aided Des. 30(13), 983–989 (2014)

    MATH  Google Scholar 

  6. Chang, M.S., Chen, L.H., Hung, L.J., Liu, Y.Z., Rossmanith, P., Sikdar, S.: An O ∗(1.4658n)-time exact algorithm for the maximum bounded-degree-1 set problem. In: Proceedings of the 31st Workshop on Combinatorial Mathematics and Computation Theory, pp. 9–18 (2014)

  7. Chang, M.S., Chen, L.H., Hung, L.J., Rossmanith, P., Su, P.C.: Fixed parameter algorithms for vertex cover P 3. Discret. Optim. 19, 12–22 (2016)

    Article  Google Scholar 

  8. Christensen, H.I., Khan, A., Pokutta, S., Tetali, P.: Approximation and online algorithms for multidimensional bin packing: a survey. Comput. Sci. Rev. 24, 63–79 (2017)

    Article  MathSciNet  Google Scholar 

  9. Demange, M., Paschos, V.T.: On-line vertex-covering. Theor. Comput. Sci. 332, 83–108 (2005)

    Article  MathSciNet  Google Scholar 

  10. Fiat, A., Woeginger, G.: Online algorithms: the state of the art. Am. Soc. Civil Eng. 129(7), 845–856 (1998)

    MATH  Google Scholar 

  11. Fujito, T.: A unified approximation algorithm for node-deletion problems. Discret. Appl. Math. 86, 213–231 (1998)

    Article  MathSciNet  Google Scholar 

  12. Irani, S., Karlin, A.R.: Online computation. In: Approximation Algorithms for NP-Hard Problems, pp. 521–564. PWS Publishing Company MA (1996)

  13. Kardoš, F., Katrenič, J., Schiermeyer, I.: On computing the minimum 3-path vertex cover and dissociation number of graphs. Theor. Comput. Sci. 412(50), 7009–7017 (2011)

    Article  MathSciNet  Google Scholar 

  14. Katrenič, J.: A fast FPT algorithm for 3-path vertex cover. Inf. Process. Lett. 116(4), 273–278 (2016)

    Article  MathSciNet  Google Scholar 

  15. Li, Y., Tu, J.: A 2-approximation algorithm for the vertex cover P 4 problem in cubic graphs. Int. J. Comput. Math. 91(10), 2103–2108 (2014)

    Article  MathSciNet  Google Scholar 

  16. Mcmahan, H.B.: A survey of algorithms and analysis for adaptive online learning. Eprint Arxiv, pp. 61–66 (2015)

  17. Novotný, M.: Design and analysis of a generalized canvas protocol. In: Proceedings of WISTP 2010 LNCS 6033, pp. 106–121 (2010)

    Chapter  Google Scholar 

  18. Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)

    Article  MathSciNet  Google Scholar 

  19. Tu, J.H.: A fixed-parameter algorithm for the vertex cover P 3 problem. Inf. Process. Lett. 115(2), 96–99 (2015)

    Article  Google Scholar 

  20. Tu, J.H., Yang, F.: The vertex cover P 3 problem in cubic graphs. Inf. Process. Lett. 113(13), 481–485 (2013)

    Article  Google Scholar 

  21. Tu, J.H., Zhou, W.L.: A primal-dual approximation algorithm for the vertex cover P 3 problem. Theor. Comput. Sci. 412(50), 7044–7048 (2011)

    Article  Google Scholar 

  22. Tu, J.H., Zhou, W.L.: A factor 2 approximation algorithm for the vertex cover P 3 problem. Inf. Process. Lett. 111(14), 683–686 (2011)

    Article  Google Scholar 

  23. Tu, J.H., Wu, L.D., Yuan, J., Cui, L.: On the vertex cover P 3 problem parameterized by treewidth. J. Comb. Optim. 34(2), 414–425 (2017)

    Article  MathSciNet  Google Scholar 

  24. Xiao, M.Y., Kou, S.W.: Kernalization and parameterized algorithms for 3-path vertex cover. In: Theory and Applications of Models of Computation LNCS 10185, pp. 654–668 (2017)

    Google Scholar 

Download references

Acknowledgements

This research is supported by NSFC (11771013, 61751303, 11531011), the Zhejiang Provincial Natural Science Foundation of China (LD19A010001, LY19A010018), and the Fundamental Research Funds for the Central Universities (No. lzujbky-2017-163).

Author information

Authors and Affiliations

  1. College of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Yubai Zhang

  2. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, Zhejiang, 321004, People’s Republic of China

    Zhao Zhang

  3. Institute of Information Science, Academia Sinica, Taibei, 11529, Taiwan

    Yishuo Shi

  4. School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

    Xianyue Li

Authors
  1. Yubai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yishuo Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xianyue Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao Zhang.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Zhang, Z., Shi, Y. et al. Algorithm for Online 3-Path Vertex Cover. Theory Comput Syst 64, 327–338 (2020). https://doi.org/10.1007/s00224-019-09944-w

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-019-09944-w

Keywords

Search

Navigation