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Approximation algorithms for minimum weight connected 3-path vertex cover

Author

Listed:
  • Ran, Yingli
  • Zhang, Zhao
  • Huang, Xiaohui
  • Li, Xiaosong
  • Du, Ding-Zhu

Abstract

A k-path vertex cover (VCPk) is a vertex set C of graph G such that every path of G on k vertices has at least one vertex in C. Because of its background in keeping data integrality of a network, minimum VCPk problem (MinVCPk) has attracted a lot of researches in recent years. This paper studies the minimum weight connected VCPk problem (MinWCVCPk), in which every vertex has a weight and the VCPk found by the algorithm induces a connected subgraph and has the minimum weight. It is known that MinWCVCPk is set-cover-hard. We present two polynomial-time approximation algorithms for MinWCVCP3. The first one is a greedy algorithm achieving approximation ratio 3ln n. The difficulty lies in its analysis dealing with a non-submodular potential function. The second algorithm is a 2-stage one, finding a VCP3 in the first stage and then adding more vertices for connection. We show that its approximation ratio is at most lnδmax+4+ln2, where δmax is the maximum degree of the graph. Considering the inapproximability of this problem, this ratio is asymptotically tight.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:347:y:2019:i:c:p:723-733
    DOI: 10.1016/j.amc.2018.11.045
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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. 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.
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