### Abstract

We consider the problem of Single-Tiered Relay Placement with Basestations, which takes as input a set (Formula presented.) of sensors and a set (Formula presented.) of basestations described as points in a normed space (Formula presented.) , and real numbers (Formula presented.). The objective is to place a minimum cardinality set (Formula presented.) of wireless relay nodes that connects (Formula presented.) and (Formula presented.) according to the following rules. The sensors in (Formula presented.) can communicate within distance (Formula presented.) , relay nodes in (Formula presented.) can communicate within distance (Formula presented.) , and basestations are considered to have an infinite broadcast range. Together the sets (Formula presented.) , and (Formula presented.) induce an undirected graph (Formula presented.) defined as follows: (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.) and (Formula presented.). Then (Formula presented.) connects (Formula presented.) and (Formula presented.) when this induced graph is connected. In the case of the two-dimensional Euclidean plane, we get a (Formula presented.) -approximation algorithm, improving the previous best ratio of 3.11. Let (Formula presented.) be the maximum number of points on a unit ball with pairwise distance strictly bigger than 1. Under certain assumptions, we have a (Formula presented.) -approximation algorithm. When biconnectivity is required, we show that a variant of our previously proposed algorithm has approximation ratio of (Formula presented.). In the case of the two-dimensional Euclidean plane, our ratio of 7 improves our previous bound of 16.

Original language | English (US) |
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Pages (from-to) | 1280-1297 |

Number of pages | 18 |

Journal | Journal of Combinatorial Optimization |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2016 |

### Keywords

- Approximation algorithm
- Biconnectivity
- Steiner points
- Wireless network

### ASJC Scopus subject areas

- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Control and Optimization
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Journal of Combinatorial Optimization*,

*31*(3), 1280-1297. https://doi.org/10.1007/s10878-014-9823-0