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Algorithmica

, Volume 80, Issue 11, pp 3270–3292 | Cite as

Approximation Algorithms for Highly Connected Multi-dominating Sets in Unit Disk Graphs

  • Takuro Fukunaga
Article

Abstract

Given an undirected graph on a node set V and positive integers k and m, a k-connected m-dominating set ((km)-CDS) is defined as a subset S of V such that each node in \(V{\setminus }S\) has at least m neighbors in S, and a k-connected subgraph is induced by S. The weighted (km)-CDS problem is to find a minimum weight (km)-CDS in a given node-weighted graph. The problem is called the unweighted (km)-CDS problem if the objective is to minimize the cardinality of a (km)-CDS. These problems have been actively studied for unit disk graphs, motivated by the application of constructing a virtual backbone in a wireless ad hoc network. In this paper, we consider the case in which \(k \le m\), and we present a simple \(O(k5^k)\)-approximation algorithm for the unweighted (km)-CDS problem, and a primal-dual \(O(k^2 \log k)\)-approximation algorithm for the weighted (km)-CDS problem.

Keywords

Connected dominating set Unit disk graph Approximation algorithm 

Notes

Acknowledgements

The author thanks anonymous referees for their careful reading and many useful comments. This work was supported by JST ERATO Grant No. JPMJER1201, and JSPS KAKENHI Grant No. 17K00040.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.National Institute of InformaticsTokyoJapan
  2. 2.JST, ERATO, Kawarabayashi Large Graph ProjectTokyoJapan

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