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# Approximating k-Connected m-Dominating Sets

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LIPIcs.ESA.2020.73.pdf
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## Acknowledgements

I thank an anonymous referee for many useful comments.

## Cite As

Zeev Nutov. Approximating k-Connected m-Dominating Sets. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 73:1-73:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.73

## Abstract

A subset S of nodes in a graph G is a k-connected m-dominating set ((k,m)-cds) if the subgraph G[S] induced by S is k-connected and every v ∈ V⧵S has at least m neighbors in S. In the k-Connected m-Dominating Set ((k,m)-CDS) problem the goal is to find a minimum weight (k,m)-cds in a node-weighted graph. For m ≥ k we obtain the following approximation ratios. For general graphs our ratio O(k ln n) improves the previous best ratio O(k² ln n) of [Z. Nutov, 2018] and matches the best known ratio for unit weights of [Z. Zhang et al., 2018]. For unit disk graphs we improve the ratio O(k ln k) of [Z. Nutov, 2018] to min{m/(m-k),k^{2/3}} ⋅ O(ln² k) - this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln² k)/ε when m ≥ (1+ε)k; furthermore, we obtain ratio min{m/(m-k), √k} ⋅ O(ln² k) for uniform weights. These results are obtained by showing the same ratios for the Subset k-Connectivity problem when the set of terminals is an m-dominating set.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• k-connected graph
• m-dominating set
• approximation algorithm
• rooted subset k-connectivity
• subset k-connectivity

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