In the Euclidean $k$-Means problem we are given a collection of $n$ points $D$ in an Euclidean space and a positive integer $k$. Our goal is to identify a collection of $k$ points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in $D$ and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual $6.357$ approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved $6.12903$ approximation. As a related result, we also show that the mentioned LP has integrality gap at least $\frac{16+\sqrt{5}}{15}>1.2157$.

中文翻译：

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欧几里得 k 均值的精细逼近

在欧几里得 $k$-Means 问题中，我们给出了欧几里德空间中 $n$ 个点 $D$ 和一个正整数 $k$ 的集合。我们的目标是在同一空间（中心）中识别一组 $k$ 点，以最小化 $D$ 中每个点与最近中心之间的欧几里德距离平方和。已知这个问题是 APX 难的，当前的最佳近似比率是基于该问题的标准 LP [Ahmadian et al. 2017] 的原始对 6.357 美元近似值。FOCS'17，SICOMP'20]。在本说明中，我们展示了对 Ahmadian 等人的分析的微小修改如何导致 $6.12903$ 近似值略有改进。作为相关结果，我们还表明，上述 LP 的完整性差距至少为 $\frac{16+\sqrt{5}}{15}>1.2157$。