## 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(k^{2} ln n) of [26] and matches the best known ratio for unit weights of [34]. For unit disk graphs we improve the ratio O(k ln k) of [26] to min { _{m}^{m}_{−}k, k^{2}/^{3} · O(ln^{2} k) – this is the first sublinear ratio for the problem, and the first polylogarithmic ratio O(ln^{2} k)/ when m ≥ (1 + )k; furthermore, we obtain ratio min {_{m}^{m}_{−}k, ^{√}k · O(ln^{2} 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.

Original language | English |
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Title of host publication | 28th Annual European Symposium on Algorithms, ESA 2020 |

Editors | Fabrizio Grandoni, Grzegorz Herman, Peter Sanders |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771627 |

DOIs | |

State | Published - 1 Aug 2020 |

Event | 28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: 7 Sep 2020 → 9 Sep 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 173 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 28th Annual European Symposium on Algorithms, ESA 2020 |
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Country/Territory | Italy |

City | Virtual, Pisa |

Period | 7/09/20 → 9/09/20 |

### Bibliographical note

Publisher Copyright:© Zeev Nutov.

## Keywords

- Approximation algorithm
- M-dominating set
- Rooted subset k-connectivity
- Subset k-connectivity
- k-connected graph