Theory of Computing Systems

, Volume 64, Issue 2, pp 327–338 | Cite as

Algorithm for Online 3-Path Vertex Cover

  • Yubai Zhang
  • Zhao ZhangEmail author
  • Yishuo Shi
  • Xianyue Li


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.


Online algorithm 3-path vertex cover 



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).


  1. 1.
    Albers, S.: Online algorithms: a survey. Math. Program. 97(1–2), 3–26 (2003)MathSciNetCrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 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)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brešar, B., Kardoš, F., Katrenič, J., Semanišin, G.: Minimum k-path vertex cover. Discret. Appl. Math. 159(12), 1189–1195 (2011)MathSciNetCrossRefGoogle Scholar
  5. 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)zbMATHGoogle Scholar
  6. 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)Google Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Demange, M., Paschos, V.T.: On-line vertex-covering. Theor. Comput. Sci. 332, 83–108 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fiat, A., Woeginger, G.: Online algorithms: the state of the art. Am. Soc. Civil Eng. 129(7), 845–856 (1998)zbMATHGoogle Scholar
  11. 11.
    Fujito, T.: A unified approximation algorithm for node-deletion problems. Discret. Appl. Math. 86, 213–231 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Irani, S., Karlin, A.R.: Online computation. In: Approximation Algorithms for NP-Hard Problems, pp. 521–564. PWS Publishing Company MA (1996)Google Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Katrenič, J.: A fast FPT algorithm for 3-path vertex cover. Inf. Process. Lett. 116(4), 273–278 (2016)MathSciNetCrossRefGoogle Scholar
  15. 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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mcmahan, H.B.: A survey of algorithms and analysis for adaptive online learning. Eprint Arxiv, pp. 61–66 (2015)Google Scholar
  17. 17.
    Novotný, M.: Design and analysis of a generalized canvas protocol. In: Proceedings of WISTP 2010 LNCS 6033, pp. 106–121 (2010)CrossRefGoogle Scholar
  18. 18.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tu, J.H.: A fixed-parameter algorithm for the vertex cover P 3 problem. Inf. Process. Lett. 115(2), 96–99 (2015)CrossRefGoogle Scholar
  20. 20.
    Tu, J.H., Yang, F.: The vertex cover P 3 problem in cubic graphs. Inf. Process. Lett. 113(13), 481–485 (2013)CrossRefGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 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)CrossRefGoogle Scholar
  23. 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)MathSciNetCrossRefGoogle Scholar
  24. 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

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Yubai Zhang
    • 1
  • Zhao Zhang
    • 2
    Email author
  • Yishuo Shi
    • 3
  • Xianyue Li
    • 4
  1. 1.College of MathematicsEast China University of Science and TechnologyShanghaiPeople’s Republic of China
  2. 2.College of Mathematics and Computer ScienceZhejiang Normal UniversityJinhuaPeople’s Republic of China
  3. 3.Institute of Information ScienceAcademia SinicaTaibeiTaiwan
  4. 4.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China

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