Facility Location on High-dimensional Euclidean Spaces

TL;DR

This paper advances the understanding of facility location problems in high-dimensional Euclidean spaces by providing improved approximation algorithms and establishing a separation from general metric spaces.

Contribution

It introduces new bifactor approximation algorithms for Euclidean Uncapacitated Facility Location, surpassing previous results and demonstrating a fundamental difference from general metrics.

Findings

First separation between Euclidean and general metric UFL.

Improved bifactor approximation algorithms for Euclidean UFL.

Enhanced understanding of approximation limits in high-dimensional spaces.

Abstract

Recent years have seen great progress in the approximability of fundamental clustering and facility location problems on high-dimensional Euclidean spaces, including $k$-Means and $k$-Median. While they admit strictly better approximation ratios than their general metric versions, their approximation ratios are still higher than the hardness ratios for general metrics, leaving the possibility that the ultimate optimal approximation ratios will be the same between Euclidean and general metrics. Moreover, such an improved algorithm for Euclidean spaces is not known for Uncapaciated Facility Location (UFL), another fundamental problem in the area. In this paper, we prove that for any $\gamma \geq 1.6774$ there exists $\varepsilon > 0$ such that Euclidean UFL admits a $(\gamma, 1 + 2e^{-\gamma} - \varepsilon)$-bifactor approximation algorithm, improving the result of Byrka and Aardal. Together with the $(\gamma, 1 + 2e^{-\gamma})$ NP-hardness in general metrics, it shows the first separation between general and Euclidean metrics for the aforementioned basic problems. We also present an $(\alpha_{Li} - \varepsilon)$-(unifactor) approximation algorithm for UFL for some $\varepsilon > 0$ in Euclidean spaces, where $\alpha_{Li} \approx 1.488$ is the best-known approximation ratio for UFL by Li.