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

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## Abstract

Given an undirected graph on a node set *V* and positive integers *k* and *m*, a *k*-connected *m*-dominating set ((*k*, *m*)-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 (*k*, *m*)-CDS problem is to find a minimum weight (*k*, *m*)-CDS in a given node-weighted graph. The problem is called the unweighted (*k*, *m*)-CDS problem if the objective is to minimize the cardinality of a (*k*, *m*)-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 (*k*, *m*)-CDS problem, and a primal-dual \(O(k^2 \log k)\)-approximation algorithm for the weighted (*k*, *m*)-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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