## Abstract

In this paper, we consider the following red-blue median problem which is a generalization of the well-studied k-median problem. The input consists of a set of red facilities, a set of blue facilities, and a set of clients in a metric space and two integers k_{r},k_{b} ≥0. The problem is to open at most k_{r} red facilities and at most k_{b} blue facilities and minimize the sum of distances of clients to their respective closest open facilities. We show, somewhat surprisingly, that the following simple local search algorithm yields a constant factor approximation for this problem. Start by opening any k_{r} red and k_{b} blue facilities. While possible, decrease the cost of the solution by closing a pair of red and blue facilities and opening a pair of red and blue facilities. We also improve the approximation factor for the prize-collecting k-median problem from 4 (Charikar et al. in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 642-641, 2001) to 3+ϵ, which matches the current best approximation factor for the k-median problem.

Original language | English |
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Pages (from-to) | 795-814 |

Number of pages | 20 |

Journal | Algorithmica |

Volume | 63 |

Issue number | 4 |

DOIs | |

State | Published - 1 Aug 2012 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© Springer Science+Business Media, LLC 2011.

## Keywords

- Facility location
- Local search algorithms
- Prize-collecting
- k-median