## Abstract

The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars.

Original language | English |
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Pages (from-to) | 18-27 |

Number of pages | 10 |

Journal | Theoretical Computer Science |

Volume | 842 |

DOIs | |

Publication status | Published - 2020 Nov 24 |

## Keywords

- Facility location
- Graph algorithm
- Multi-service location

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)