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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Björklund A, Husfeldt T, Kaski P, Koivisto AM. Narrow sieves for parameterized paths and packings, arXiv:1007.1161
Brešar B, Kardoš F, Katrenič J, Semaniš G (2011) Minimum \(k\)-path vertex cover. Discrete Appl Math 159:1189–1195
Chang M, Chen L, Hung L, Rossmanith P, Su P (2016) Fixed-parameter algorithms for vertex cover \(P_3\). Discrete Optim 19:12–22
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):202–208
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):321–333
Hochbaum DS, Maass W (1985) Approximation schemes for covering and packing problems in image processing and VLSI. J ACM 32:130–136
Li X, Zhang Z, Huang X (2016) Approximation algorithms for minimum (weight) connected \(k\)-path vertex cover. Discrete Appl Math 205:101–108
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):1250014
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 56:449–458
Novotny M (2010) Design and analysis of a generalized canvas protocol. In: Proceedings of WISTP 2010, in: LNCS 6033:106–121
Oler N (1961) An inequality in the geometry of numbers. Acta Math. 105:19–48
Tu J, Zhou W (2011) A factor 2 approximation algorithm for the vertex cover \(P_{3}\) problem. Inf. Process. Lett. 111:683–686
Tu J, Zhou W (2011) A primal-dual approximation algorithm for the vertex cover \(P_3\) problem. Theor Comput Sci 412:7044–7048
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):283–289
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 571:58–66
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 33:106–122
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):451–458
Zhang Z, Li X, Shi Y, Nie H, Zhu Y (2017) PTAS for minimum \(k\)-path vertex cover in ball graph. Inf Process Lett 119:9–13
Acknowledgements
This research is supported by NSFC (11771013, 11531011, 61502431).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chen, L., Huang, X. & Zhang, Z. A simpler PTAS for connected k-path vertex cover in homogeneous wireless sensor network. J Comb Optim 36, 35–43 (2018). https://doi.org/10.1007/s10878-018-0283-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-018-0283-9