In the (continuous) Euclidean $k$-center problem, given $n$ points in $\mathbb{R}^d$ and an integer $k$, the goal is to find $k$ center points in $\mathbb{R}^d$ that minimize the maximum Euclidean distance from any input point to its closest center. In this paper, we establish conditional lower bounds for this problem in constant dimensions in two settings. $\bullet$ Parameterized by $k$: Assuming the Exponential Time Hypothesis (ETH), we show that there is no $f(k)n^{o(k^{1-1/d})}$-time algorithm for the Euclidean $k$-center problem. This result shows that the algorithm of Agarwal and Procopiuc [SODA 1998; Algorithmica 2002] is essentially optimal. Furthermore, our lower bound rules out any $(1+\varepsilon)$-approximation algorithm running in time $(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$, thereby establishing near-optimality of the corresponding approximation scheme by the same authors. $\bullet$ Small $k$: Assuming the 3-SUM hypothesis, we prove that for any $\varepsilon>0$ there is no $O(n^{2-\varepsilon})$-time algorithm for the Euclidean $2$-center problem in $\mathbb{R}^3$. This settles an open question posed by Agarwal, Ben Avraham, and Sharir [SoCG 2010; Computational Geometry 2013]. In addition, under the same hypothesis, we prove that for any $\varepsilon > 0$, the Euclidean $6$-center problem in $\mathbb{R}^2$ also admits no $O(n^{2-\varepsilon})$-time algorithm. The technical core of all our proofs is a novel geometric embedding of a system of linear equations. We construct a point set where each variable corresponds to a specific collection of points, and the geometric structure ensures that a small-radius clustering is possible if and only if the system has a valid solution.

翻译：在（连续）欧几里得$k$-中心问题中，给定$\mathbb{R}^d$中的$n$个点和一个整数$k$，目标是找到$\mathbb{R}^d$中的$k$个中心点，使得任意输入点到其最近中心的最大欧几里得距离最小化。在本文中，我们在两个场景下为该问题在常数维度中建立了条件下界。$\bullet$ 参数化于$k$：假设指数时间假说（ETH），我们证明不存在$f(k)n^{o(k^{1-1/d})}$时间的算法用于欧几里得$k$-中心问题。该结果表明Agarwal和Procopiuc的算法[SODA 1998; Algorithmica 2002]本质上是最优的。此外，我们的下界排除了任何运行时间为$(k/\varepsilon)^{o(k^{1-1/d})}n^{O(1)}$的$(1+\varepsilon)$近似算法，从而确立了同一作者相应近似方案近乎最优性。$\bullet$ 小$k$：假设3-SUM假说，我们证明对于任意$\varepsilon>0$，在$\mathbb{R}^3$中不存在$O(n^{2-\varepsilon})$时间的算法用于欧几里得2-中心问题。这解决了Agarwal、Ben Avraham和Sharir提出的一个开放问题[SoCG 2010; Computational Geometry 2013]。此外，在相同假说下，我们证明对于任意$\varepsilon>0$，$\mathbb{R}^2$中的欧几里得6-中心问题也不存在$O(n^{2-\varepsilon})$时间的算法。我们所有证明的技术核心是一个新颖的线性方程组的几何嵌入。我们构造了一个点集，其中每个变量对应于一个特定的点集合，而几何结构确保只有当系统有有效解时，才能实现小半径的聚类。