A simpler PTAS for connected k-path vertex cover in homogeneous wireless sensor network

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Abstract

Because of its application in the field of security in wireless sensor networks, k-path vertex cover (\(\hbox {VCP}_k\)) has received a lot of attention in recent years. Given a graph \(G=(V,E)\), a vertex set \(C\subseteq V\) is a k-path vertex cover (\(\hbox {VCP}_k\)) of G if every path on k vertices has at least one vertex in C, and C is a connected k-path vertex cover of G (\(\hbox {CVCP}_k\)) if furthermore the subgraph of G induced by C is connected. A homogeneous wireless sensor network can be modeled as a unit disk graph. This paper presents a new PTAS for \(\hbox {MinCVCP}_k\) on unit disk graphs. Compared with previous PTAS given by Liu et al., our method not only simplifies the algorithm and reduces the time-complexity, but also simplifies the analysis by a large amount.

Keywords

Connected k-path vertex cover Unit disk graph PTAS Approximation algorithm 

Notes

Acknowledgements

This research is supported by NSFC (11771013, 11531011, 61502431).

References

  1. Björklund A, Husfeldt T, Kaski P, Koivisto AM. Narrow sieves for parameterized paths and packings, arXiv:1007.1161
  2. Brešar B, Kardoš F, Katrenič J, Semaniš G (2011) Minimum \(k\)-path vertex cover. Discrete Appl Math 159:1189–1195MathSciNetCrossRefMATHGoogle Scholar
  3. Chang M, Chen L, Hung L, Rossmanith P, Su P (2016) Fixed-parameter algorithms for vertex cover \(P_3\). Discrete Optim 19:12–22MathSciNetCrossRefGoogle Scholar
  4. Cheng X, Huang X, Li D, Wu W, Du D (2003) A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks 42(4):202–208MathSciNetCrossRefMATHGoogle Scholar
  5. Gao X, Wang W, Zhang Z, Zhu S, Wu W (2010) A PTAS for minimum \(d\)-hop connected dominating set in growth-bounded graphs. Optim Lett 4(3):321–333MathSciNetCrossRefMATHGoogle Scholar
  6. Hochbaum DS, Maass W (1985) Approximation schemes for covering and packing problems in image processing and VLSI. J ACM 32:130–136MathSciNetCrossRefMATHGoogle Scholar
  7. Li X, Zhang Z, Huang X (2016) Approximation algorithms for minimum (weight) connected \(k\)-path vertex cover. Discrete Appl Math 205:101–108MathSciNetCrossRefMATHGoogle Scholar
  8. Liu Q, Li X, Wu L, Du H, Zhang Z, Wu W, Hu X, Xu Y (2012) A new proof for Zassenhaus–Groemer–Oler inequality. Discrete Math Algorithms Appl 4(2):1250014MathSciNetCrossRefMATHGoogle Scholar
  9. Liu X, Lu H, Wang W, Wu W (2013) PTAS for the minimum \(k\)-path connected vertex cover problem in unit disk graphs. J Glob Optim 56:449–458MathSciNetCrossRefMATHGoogle Scholar
  10. Novotny M (2010) Design and analysis of a generalized canvas protocol. In: Proceedings of WISTP 2010, in: LNCS 6033:106–121Google Scholar
  11. Oler N (1961) An inequality in the geometry of numbers. Acta Math. 105:19–48MathSciNetCrossRefMATHGoogle Scholar
  12. Tu J, Zhou W (2011) A factor 2 approximation algorithm for the vertex cover \(P_{3}\) problem. Inf. Process. Lett. 111:683–686CrossRefMATHGoogle Scholar
  13. Tu J, Zhou W (2011) A primal-dual approximation algorithm for the vertex cover \(P_3\) problem. Theor Comput Sci 412:7044–7048CrossRefMATHGoogle Scholar
  14. Wang W, Kim D, Sohaee N, Ma C, Wu W (2009) A PTAS for minimum \(d\)-hop underwater sink placement problem in 2-D underwater sensor networks. Discrete Math Algorithms Appl 1(2):283–289MathSciNetCrossRefMATHGoogle Scholar
  15. Wang L, Zhang X, Zhang Z, Broersma H (2015) A PTAS for the minimum weight connected vertex cover \(P_3\) problem on unit disk graphs. Theor Comput Sci 571:58–66CrossRefMATHGoogle Scholar
  16. Wang L, Du W, Zhang Z, Zhang X (2017) A PTAS for minimum weighted connected vertex cover \(P_3\) problem in 3-dimensional wireless sensor networks. J Comb Optim 33:106–122MathSciNetCrossRefMATHGoogle Scholar
  17. Zhang Z, Gao X, Wu W, Du D (2009) A PTAS for minimum connected dominating set in 3-dimensional wireless sensor networks. J Glob Optim 45(3):451–458MathSciNetCrossRefMATHGoogle Scholar
  18. Zhang Z, Li X, Shi Y, Nie H, Zhu Y (2017) PTAS for minimum \(k\)-path vertex cover in ball graph. Inf Process Lett 119:9–13MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics Physics and Information EngineeringZhejiang Normal UniversityJinhuaChina
  2. 2.Library and Information CenterZhejiang Normal UniversityJinhuaChina

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