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原文摘要

In the Euclidean $k$-means problem, we are given a set of $n$ points in $\mathbb{R}^d$, and the goal is to find $k$ representative points in space so as to minimize the sum of the squared distances from each point to its closest representative. Algorithmically, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] obtain a $(1+\varepsilon)$-approximation for high-dimensional Euclidean $k$-means in time $2^{(k/\varepsilon)^{O(1)}} \cdot dn^{O(1)}$, which was subsequently improved to $2^{\tilde{O}(k/\varepsilon)} + O(ndk)$ by Feldman, Monemizadeh, and Sohler [SoCG'07]. On the lower-bound side, a recent work of Cohen-Addad, Karthik, Saulpic, and Schwiegelshohn [SoCG'26] proposed the Exponential Time for Expanders Hypothesis (XXH), which asserts exponential-time hardness for $(1+\varepsilon)$-approximating Vertex Cover on vertex expanders, and used it to prove nearly matching running time lower bound whenever $k=\tildeω(1/\varepsilon)$. We observe that XXH does not hold in the parameter regime needed for the above result, and show how to repair their reduction under a more standard assumption. More precisely, we show that assuming the Almost Exponential Time Hypothesis for 3-SAT (Almost ETH), for every constant $β\in(0,1)$, Euclidean $k$-means in $\mathbb{R}^d$ has no $(1+\varepsilon)$-approximation algorithm running in time $2^{(k/\varepsilon)^{1-β}}\text{poly}(n,d)$, whenever $k=\tildeω(1/\varepsilon)$. By modifying the construction, we also obtain a similar result for approximating the Euclidean $k$-median problem.