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Breaking the O (ln n ) Barrier: An Enhanced Approximation Algorithm for Fault-Tolerant Minimum Weight Connected Dominating Set

Author

Listed:
  • Jiao Zhou

    (College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang Province 321004, China)

  • Zhao Zhang

    (College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang Province 321004, China)

  • Shaojie Tang

    (Naveen Jindal School of Management, The University of Texas at Dallas, Richardson, Texas 75080)

  • Xiaohui Huang

    (College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang Province 321004, China)

  • Ding-Zhu Du

    (Department of Computer Science, University of Texas at Dallas, Richardson, Texas 75080)

Abstract

Finding a connected dominating set (CDS) in a given graph is a fundamental problem and has been studied intensively for a long time because of its application in computer science and operations research, e.g., connected facility location and wireless networks. In some cases, fault-tolerance is desirable. Taking wireless networks as an example, since wireless nodes may fail due to accidental damage or energy depletion, it is desirable that the virtual backbone has some fault-tolerance. Such a problem can be modeled as finding a minimum k -connected m -fold dominating set (( k , m )-CDS) of a graph G = ( V , E ), which is a node set D such that every node outside of D has at least m neighbors in D and the subgraph of G induced by D is k -connected. In this paper, we study the minimum weight (1, m )-CDS problem ((1, m )-MWCDS), and present an ( H ( δ + m ) + 2 H ( δ − 1))-approximation algorithm, where δ is the maximum degree of the graph and H (·) is the Harmonic number. Notice that the state-of-the-art algorithm achieves O (ln( n ))-approximation factor for the (1, 1)-MWCDS problem, where n is the number of nodes. Our work improves this ratio to O (ln δ ) for an even more general problem: (1, m )-MWCDS. Such an improvement also enables us to obtain a (3.67 + α )-approximation for the (1, m )-MWCDS problem on unit disk graph, where α is the performance ratio for the minimum weight m -fold dominating set problem on unit disk graph.

Suggested Citation

  • Jiao Zhou & Zhao Zhang & Shaojie Tang & Xiaohui Huang & Ding-Zhu Du, 2018. "Breaking the O (ln n ) Barrier: An Enhanced Approximation Algorithm for Fault-Tolerant Minimum Weight Connected Dominating Set," INFORMS Journal on Computing, INFORMS, vol. 30(2), pages 225-235, May.
    • Handle: RePEc:inm:orijoc:v:30:y:2018:i:2:p:225-235
    DOI: 10.1287/ijoc.2017.0775
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    References listed on IDEAS

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    1. Yaochun Huang & Xiaofeng Gao & Zhao Zhang & Weili Wu, 2009. "A better constant-factor approximation for weighted dominating set in unit disk graph," Journal of Combinatorial Optimization, Springer, vol. 18(2), pages 179-194, August.
    2. Jiao Zhou & Zhao Zhang & Weili Wu & Kai Xing, 2014. "A greedy algorithm for the fault-tolerant connected dominating set in a general graph," Journal of Combinatorial Optimization, Springer, vol. 28(1), pages 310-319, July.
    3. Feng Zou & Xianyue Li & Suogang Gao & Weili Wu, 2009. "Node-weighted Steiner tree approximation in unit disk graphs," Journal of Combinatorial Optimization, Springer, vol. 18(4), pages 342-349, November.
    4. Austin Buchanan & Je Sang Sung & Sergiy Butenko & Eduardo L. Pasiliao, 2015. "An Integer Programming Approach for Fault-Tolerant Connected Dominating Sets," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 178-188, February.
    5. Bernard Gendron & Abilio Lucena & Alexandre Salles da Cunha & Luidi Simonetti, 2014. "Benders Decomposition, Branch-and-Cut, and Hybrid Algorithms for the Minimum Connected Dominating Set Problem," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 645-657, November.
    6. M. Gisela Bardossy & S. Raghavan, 2010. "Dual-Based Local Search for the Connected Facility Location and Related Problems," INFORMS Journal on Computing, INFORMS, vol. 22(4), pages 584-602, November.
    7. Weiping Shang & Frances Yao & Pengjun Wan & Xiaodong Hu, 2008. "On minimum m-connected k-dominating set problem in unit disc graphs," Journal of Combinatorial Optimization, Springer, vol. 16(2), pages 99-106, August.
    8. Yishuo Shi & Yaping Zhang & Zhao Zhang & Weili Wu, 2016. "A greedy algorithm for the minimum $$2$$ 2 -connected $$m$$ m -fold dominating set problem," Journal of Combinatorial Optimization, Springer, vol. 31(1), pages 136-151, January.
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