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On connected dominating sets of restricted diameter

Author

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  • Buchanan, Austin
  • Sung, Je Sang
  • Boginski, Vladimir
  • Butenko, Sergiy

Abstract

A connected dominating set (CDS) is commonly used to model a virtual backbone of a wireless network. To bound the distance that information must travel through the network, we explicitly restrict the diameter of a CDS to be no more than s leading to the concept of a dominating s-club. We prove that for any fixed positive integer s it is NP-complete to determine if a graph has a dominating s-club, even when the graph has diameter s+1. As a special case it is NP-complete to determine if a graph of diameter two has a dominating clique. We then propose a compact integer programming formulation for the related minimization problem, enhance the approach with variable fixing rules and valid inequalities, and present computational results.

Suggested Citation

  • Buchanan, Austin & Sung, Je Sang & Boginski, Vladimir & Butenko, Sergiy, 2014. "On connected dominating sets of restricted diameter," European Journal of Operational Research, Elsevier, vol. 236(2), pages 410-418.
    • Handle: RePEc:eee:ejores:v:236:y:2014:i:2:p:410-418
    DOI: 10.1016/j.ejor.2013.11.036
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    References listed on IDEAS

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    1. Bourjolly, Jean-Marie & Laporte, Gilbert & Pesant, Gilles, 2002. "An exact algorithm for the maximum k-club problem in an undirected graph," European Journal of Operational Research, Elsevier, vol. 138(1), pages 21-28, April.
    2. Abilio Lucena & Nelson Maculan & Luidi Simonetti, 2010. "Reformulations and solution algorithms for the maximum leaf spanning tree problem," Computational Management Science, Springer, vol. 7(3), pages 289-311, July.
    3. Robert Mokken, 1979. "Cliques, clubs and clans," Quality & Quantity: International Journal of Methodology, Springer, vol. 13(2), pages 161-173, April.
    4. Veremyev, Alexander & Boginski, Vladimir, 2012. "Identifying large robust network clusters via new compact formulations of maximum k-club problems," European Journal of Operational Research, Elsevier, vol. 218(2), pages 316-326.
    5. Balabhaskar Balasundaram & Sergiy Butenko & Svyatoslav Trukhanov, 2005. "Novel Approaches for Analyzing Biological Networks," Journal of Combinatorial Optimization, Springer, vol. 10(1), pages 23-39, August.
    6. Pattillo, Jeffrey & Youssef, Nataly & Butenko, Sergiy, 2013. "On clique relaxation models in network analysis," European Journal of Operational Research, Elsevier, vol. 226(1), pages 9-18.
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    Cited by:

    1. Hamidreza Validi & Austin Buchanan, 2020. "The Optimal Design of Low-Latency Virtual Backbones," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 952-967, October.
    2. Zeynep Ertem & Eugene Lykhovyd & Yiming Wang & Sergiy Butenko, 2020. "The maximum independent union of cliques problem: complexity and exact approaches," Journal of Global Optimization, Springer, vol. 76(3), pages 545-562, March.
    3. Li, Xiangyong & Aneja, Y.P., 2017. "Regenerator location problem: Polyhedral study and effective branch-and-cut algorithms," European Journal of Operational Research, Elsevier, vol. 257(1), pages 25-40.
    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.

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