## Abstract

In the discrete k-Center problem, we are given a metric space (P,dist) where |P| = n and the goal is to select a set C ⊆ P of k centers which minimizes the maximum distance of a point in P from its nearest center. For any ε > 0, Agarwal and Procopiuc [SODA '98, Algorithmica '02] designed an (1+ε)-approximation algorithm for this problem in d-dimensional Euclidean space which runs in O(dn log k) + (k/ε)^{O (k^{1-1/d})}⋅ n^{O(1)} time. In this paper we show that their algorithm is essentially optimal: if for some d ≥ 2 and some computable function f, there is an f(k)⋅(1/ε)^{o (k^{1-1/d})} ⋅ n^{o (k^{1-1/d})} time algorithm for (1+ε)-approximating the discrete k-Center on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails.

We obtain our lower bound by designing a gap reduction from a d-dimensional constraint satisfaction problem (CSP) to discrete d-dimensional k-Center. This reduction has the property that there is a fixed value ε (depending on the CSP) such that the optimal radius of k-Center instances corresponding to satisfiable and unsatisfiable instances of the CSP is < 1 and ≥ (1+ε) respectively. Our claimed lower bound on the running time for approximating discrete k-Center in d-dimensions then follows from the lower bound due to Marx and Sidiropoulos [SoCG '14] for checking the satisfiability of the aforementioned d-dimensional CSP.

As a byproduct of our reduction, we also obtain that the exact algorithm of Agarwal and Procopiuc [SODA '98, Algorithmica '02] which runs in n^{O (d⋅ k^{1-1/d})} time for discrete k-Center on n points in d-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some d ≥ 2 and some computable function f, there is an f(k)⋅n^{o (k^{1-1/d})} time exact algorithm for the discrete k-Center problem on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. Previously, such a lower bound was only known for d = 2 and was implicit in the work of Marx [IWPEC '06].

We obtain our lower bound by designing a gap reduction from a d-dimensional constraint satisfaction problem (CSP) to discrete d-dimensional k-Center. This reduction has the property that there is a fixed value ε (depending on the CSP) such that the optimal radius of k-Center instances corresponding to satisfiable and unsatisfiable instances of the CSP is < 1 and ≥ (1+ε) respectively. Our claimed lower bound on the running time for approximating discrete k-Center in d-dimensions then follows from the lower bound due to Marx and Sidiropoulos [SoCG '14] for checking the satisfiability of the aforementioned d-dimensional CSP.

As a byproduct of our reduction, we also obtain that the exact algorithm of Agarwal and Procopiuc [SODA '98, Algorithmica '02] which runs in n^{O (d⋅ k^{1-1/d})} time for discrete k-Center on n points in d-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some d ≥ 2 and some computable function f, there is an f(k)⋅n^{o (k^{1-1/d})} time exact algorithm for the discrete k-Center problem on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. Previously, such a lower bound was only known for d = 2 and was implicit in the work of Marx [IWPEC '06].

Original language | English |
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Title of host publication | 38th International Symposium on Computational Geometry (SoCG 2022) |

Publisher | Schloss Dagstuhl |

Pages | 1-15 |

Number of pages | 15 |

ISBN (Electronic) | 9783959772273 |

DOIs | |

Publication status | Published - 1 Jun 2022 |

Event | 38th International Symposium on Computational Geometry (SoCG 2022) - Berlin, Germany Duration: 7 Jun 2022 → 10 Jun 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics (LIPIcs) |
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Publisher | Schloss Dagstuhl |

Volume | 224 |

ISSN (Electronic) | 1868-8969 |

### Conference

Conference | 38th International Symposium on Computational Geometry (SoCG 2022) |
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Abbreviated title | SoCG 2022 |

Country/Territory | Germany |

City | Berlin |

Period | 7/06/22 → 10/06/22 |

## Keywords

- k-center
- Euclidean space
- Exponential Time Hypothesis (ETH)
- lower bound

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