## Abstract

The input to the line-sets k-median problem is an integer k ≥ 1, and a set L = {L_{1},..., L_{n}} that contains n sets of lines in R^{d}. The goal is to compute a set C of k centers (points in R^{d}) that minimizes the sum Σ_{L∈L} min_{ℓ}∈L,c∈_{C} dist(ℓ, c) of Euclidean distances from each set to its closest center, where dist(ℓ, c):= min_{x}∈_{ℓ} ∥x - c∥_{2}. An ε-coreset for this problem is a weighted subset of sets in L that approximates this sum up to 1 ± ε multiplicative factor, for every set C of k centers. We prove that every such input set L has a small ε-coreset, and provide the first coreset construction for this problem and its variants. The coreset consists of O(log^{2} n) weighted line-sets from L, and is constructed in O(n log n) time for every fixed d, k ≥ 1 and ε ∈ (0, 1). The main technique is based on a novel reduction to a “fair clustering” of colored points to colored centers. We then provide a coreset for this coloring problem, which may be of independent interest. Open source code and experiments are also provided.

Original language | English |
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Title of host publication | Advances in Neural Information Processing Systems 35 - 36th Conference on Neural Information Processing Systems, NeurIPS 2022 |

Editors | S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, A. Oh |

Publisher | Neural information processing systems foundation |

ISBN (Electronic) | 9781713871088 |

State | Published - 2022 |

Event | 36th Conference on Neural Information Processing Systems, NeurIPS 2022 - New Orleans, United States Duration: 28 Nov 2022 → 9 Dec 2022 |

### Publication series

Name | Advances in Neural Information Processing Systems |
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Volume | 35 |

ISSN (Print) | 1049-5258 |

### Conference

Conference | 36th Conference on Neural Information Processing Systems, NeurIPS 2022 |
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Country/Territory | United States |

City | New Orleans |

Period | 28/11/22 → 9/12/22 |

### Bibliographical note

Publisher Copyright:© 2022 Neural information processing systems foundation. All rights reserved.

## ASJC Scopus subject areas

- Computer Networks and Communications
- Information Systems
- Signal Processing