Facility Location on High-dimensional Euclidean Spaces

TL;DR

This work studies Uncapacitated Facility Location (UFL) in high-dimensional Euclidean spaces and establishes a first strict separation from general metrics by proving a bifactor approximation $(\gamma, 1+2e^{-\gamma}-\varepsilon)$ for Euclidean UFL when $\gamma\ge 1.6774$ and a unifactor approximation $(\alpha_{\mathrm{Li}}-\varepsilon)$, leveraging Euclidean geometry to surpass the Byrka–Aardal baseline. The authors extend Li’s framework, introduce geometry-driven clustering and interval-based analyses, and integrate JMS for facility-dominant cases to achieve tight bounds; they also prove Euclidean UFL is APX-hard. The approach combines a refined LP-rounding scheme with geometric arguments to reduce rerouting costs, yielding practical improvements over prior results and illuminating intrinsic differences between Euclidean and general metrics. These results motivate further exploration of geometry-aware rounding and hybrid algorithms (JMS/JV) for even tighter UFL guarantees in high dimensions.

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 $γ\geq 1.6774$ there exists $\varepsilon > 0$ such that Euclidean UFL admits a $(γ, 1 + 2e^{-γ} - \varepsilon)$-bifactor approximation algorithm, improving the result of Byrka and Aardal. Together with the $(γ, 1 + 2e^{-γ})$ NP-hardness in general metrics, it shows the first separation between general and Euclidean metrics for the aforementioned basic problems. We also present an $(α_{Li} - \varepsilon)$-(unifactor) approximation algorithm for UFL for some $\varepsilon > 0$ in Euclidean spaces, where $α_{Li} \approx 1.488$ is the best-known approximation ratio for UFL by Li.

Figures (4)

• Figure 1: A simple case when $cost_{j'}(j) \approx 3$. There is a client-dense region on the left.
• Figure 2: $\mathcal{C}_{k} \cup \mathcal{D}_{k}$ contains facilities positioned on the opposite side of $j$.
• Figure 3: Two examples in a homogeneous network where every neighbor of center $j'$ belongs to $N^-_{j'}$ without creating a dense region.
• Figure 4: Intersection between a $T_{j'j}$ and a remote arm $\mathcal{C}_j \cup \mathcal{D}_j$.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Lemma 3.1
• proof
• Lemma 3.2