## Abstract

Let R be a finite set of terminals in a convex metric space (M,d). We give approximation algorithms for problems of finding a minimum size set S⊆M of additional points such that the unit-disc graph G[R∪S] of R∪S satisfies some connectivity properties. Let Δ be the maximum number of independent points in a unit ball. For the Steiner Tree with Minimum Number of Steiner Points problem we obtain approximation ratio 1+ln(Δ−1)+ϵ which in R^{2} reduces to 1+ln4+ϵ<2.3863+ϵ; this improves the ratios ⌊(Δ+1)/2⌋+1+ϵ of [19] and 2.5+ϵ of [7], respectively. For the Steiner Forest with Minimum Number of Steiner Points problem we give a simple Δ-approximation algorithm, improving the ratio 2(Δ−1) of [21]. We also simplify the Δ-approximation of [4], when G[R∪S] should be 2-connected.

Original language | English |
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Pages (from-to) | 53-64 |

Number of pages | 12 |

Journal | Journal of Computer and System Sciences |

Volume | 98 |

DOIs | |

State | Published - Dec 2018 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier Inc.

## Keywords

- 2-connectivity
- Approximation algorithms
- Steiner forest
- Steiner tree
- Unit-disc graph
- Wireless networks