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A PTAS for minimum weighted connected vertex cover $$P_3$$ P 3 problem in 3-dimensional wireless sensor networks

Author

Listed:
  • Limin Wang

    (Nanjing Normal University
    Nanjing University)

  • Wenxue Du

    (Anhui University)

  • Zhao Zhang

    (Zhejiang Normal University)

  • Xiaoyan Zhang

    (Nanjing Normal University
    University of Twente)

Abstract

Given a connected and weighted graph $$G=(V, E)$$ G = ( V , E ) with each vertex v having a nonnegative weight w(v), the minimum weighted connected vertex cover $$P_{3}$$ P 3 problem $$(MWCVCP_{3})$$ ( M W C V C P 3 ) is required to find a subset C of vertices of the graph with minimum total weight, such that each path with length 2 has at least one vertex in C, and moreover, the induced subgraph G[C] is connected. This kind of problem has many applications concerning wireless sensor networks and ad hoc networks. When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. In this paper, we propose a new concept called weak c-local and give the first polynomial time approximation scheme (PTAS) for $$MWCVCP_{3}$$ M W C V C P 3 in unit ball graphs when the weight is smooth and weak c-local.

Suggested Citation

  • Limin Wang & Wenxue Du & Zhao Zhang & Xiaoyan Zhang, 2017. "A PTAS for minimum weighted connected vertex cover $$P_3$$ P 3 problem in 3-dimensional wireless sensor networks," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 106-122, January.
    • Handle: RePEc:spr:jcomop:v:33:y:2017:i:1:d:10.1007_s10878-015-9937-z
    DOI: 10.1007/s10878-015-9937-z
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    References listed on IDEAS

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    1. Xianliang Liu & Hongliang Lu & Wei Wang & Weili Wu, 2013. "PTAS for the minimum k-path connected vertex cover problem in unit disk graphs," Journal of Global Optimization, Springer, vol. 56(2), pages 449-458, June.
    2. Lidan Fan & Zhao Zhang & Wei Wang, 2011. "PTAS for minimum weighted connected vertex cover problem with c-local condition in unit disk graphs," Journal of Combinatorial Optimization, Springer, vol. 22(4), pages 663-673, November.
    3. Ding-Zhu Du & Ker-I Ko & Xiaodong Hu, 2012. "Design and Analysis of Approximation Algorithms," Springer Optimization and Its Applications, Springer, number 978-1-4614-1701-9, June.
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    Cited by:

    1. Lina Chen & Xiaohui Huang & Zhao Zhang, 2018. "A simpler PTAS for connected k-path vertex cover in homogeneous wireless sensor network," Journal of Combinatorial Optimization, Springer, vol. 36(1), pages 35-43, July.
    2. Raka Jovanovic & Antonio P. Sanfilippo & Stefan Voß, 2022. "Fixed set search applied to the multi-objective minimum weighted vertex cover problem," Journal of Heuristics, Springer, vol. 28(4), pages 481-508, August.
    3. Ran, Yingli & Zhang, Zhao & Huang, Xiaohui & Li, Xiaosong & Du, Ding-Zhu, 2019. "Approximation algorithms for minimum weight connected 3-path vertex cover," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 723-733.
    4. Shuli Hu & Xiaoli Wu & Huan Liu & Yiyuan Wang & Ruizhi Li & Minghao Yin, 2019. "Multi-Objective Neighborhood Search Algorithm Based on Decomposition for Multi-Objective Minimum Weighted Vertex Cover Problem," Sustainability, MDPI, vol. 11(13), pages 1-21, July.

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