## Abstract

Let R be a finite set of terminals in a metric space (M,d). We consider 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. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem G[R ∪ S] should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-MSP) problem we are given a set D ⊆ R × R of demand pairs and G[R ∪ S] should contains a uv-path for every uv ∈ D. Let Δ be the maximum number of points in a unit ball such that the distance between any two of them is larger than 1. It is known that Δ = 5 in ℝ^{2}. The previous known approximation ratio for ST-MSP was ⌊(Δ+1)/2⌋ +1+ϵ in an arbitrary normed space [15], and 2.5+ϵ in the Euclidean space ℝ^{2} [5]. Our approximation ratio for ST-MSP is 1+ln(Δ−1)+ϵ in an arbitrary normed space, which in ℝ^{2} reduces to 1+ln4+ϵ < 2.3863+ϵ. For SF-MSP we give a simple Δ- approximation algorithm, improving the folklore ratio 2(Δ−1). Finally, we generalize and simplify the Δ-approximation of Calinescu [3] for the 2-Connectivity-MSP problem, where G[R ∪ S] should be 2-connected.

Original language | English |
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Title of host publication | Approximation and Online Algorithms - 12th International Workshop, WAOA 2014, Revised Selected Papers |

Editors | Ola Svensson, Evripidis Bampis |

Publisher | Springer Verlag |

Pages | 95-106 |

Number of pages | 12 |

ISBN (Print) | 9783319182629 |

DOIs | |

State | Published - 2015 |

Event | 12th International Workshop on Approximation and Online Algorithms, WAOA 2014 - Wroclaw, Poland Duration: 11 Sep 2014 → 12 Sep 2014 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8952 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 12th International Workshop on Approximation and Online Algorithms, WAOA 2014 |
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Country/Territory | Poland |

City | Wroclaw |

Period | 11/09/14 → 12/09/14 |

### Bibliographical note

Publisher Copyright:© Springer International Publishing Switzerland 2015.

## Keywords

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