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.

翻译：近年来，高维欧几里得空间中的基础聚类与设施选址问题（包括 $k$-均值与 $k$-中值）的近似性研究取得了显著进展。尽管这些问题在欧几里得空间中的近似比严格优于其一般度量空间版本，但其近似比仍高于一般度量空间中的硬度比，这意味着欧几里得空间与一般度量空间中的最优近似比最终可能相同。此外，对于该领域的另一个基本问题——无容量限制设施选址问题，目前尚未发现针对欧几里得空间的改进算法。本文证明，对于任意 $\gamma \geq 1.6774$，存在 $\varepsilon > 0$，使得欧几里得无容量限制设施选址问题具有 $(\gamma, 1 + 2e^{-\gamma} - \varepsilon)$-双因子近似算法，改进了 Byrka 与 Aardal 的结果。结合一般度量空间中 $(\gamma, 1 + 2e^{-\gamma})$ 的 NP 困难性，这首次揭示了上述基础问题在一般度量与欧几里得度量之间的分离性。我们还针对欧几里得空间中的无容量限制设施选址问题，提出了 $(\alpha_{Li} - \varepsilon)$-（单因子）近似算法（其中 $\varepsilon > 0$，$\alpha_{Li} \approx 1.488$ 是 Li 提出的当前最优近似比）。