## Abstract

We study a continuous facility location problem on undirected graphs where all edges have unit length and where the facilities may be positioned at the vertices as well as at interior points of the edges. The goal is to cover the entire graph with a minimum number of facilities with covering range δ > 0. In other words, we want to position as few facilities as possible subject to the condition that every point on every edge is at distance at most δ from one of these facilities. We investigate this covering problem in terms of the rational parameter δ. We prove that the problem is polynomially solvable whenever δ is a unit fraction, and that the problem is NP-hard for all non unit fractions δ. We also analyze the parametrized complexity with the solution size as parameter: The resulting problem is fixed parameter tractable for all δ < 3/2, and it is W[2]-hard for all δ ≥ 3/2.

Original language | English |
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Title of host publication | Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, Proceedings |

Subtitle of host publication | 21st International Conference, IPCO 2020, London, UK, June 8–10, 2020, Proceedings |

Editors | Daniel Bienstock, Giacomo Zambelli |

Place of Publication | Cham |

Publisher | Springer |

Pages | 171-181 |

Number of pages | 11 |

ISBN (Print) | 9783030457709 |

DOIs | |

Publication status | Published - 1 Jan 2020 |

Event | 21st Conference on Integer Programming and Combinatorial Optimization: IPCO 2020 - London School of Economics, Virtuell, United Kingdom Duration: 8 Jun 2020 → 10 Jun 2020 Conference number: 21 http://www.lse.ac.uk/IPCO-2020 |

### 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 | 12125 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st Conference on Integer Programming and Combinatorial Optimization |
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Abbreviated title | IPCO 2020 LSE |

Country/Territory | United Kingdom |

City | Virtuell |

Period | 8/06/20 → 10/06/20 |

Internet address |

## Keywords

- Graph theory
- Location theory
- Parametrized complexity

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)