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 $\frac{16+\sqrt{5}}{15}>1.2157$.

翻译：在欧几里德$k$-Means问题中,我们在一个欧几里德空间收集了10美元,一个正整数美元。我们的目标是在同一空间(中间点)中确定1美元点的集合,以便最大限度地减少欧几里德每点之间以美元计的平方位距离与最接近的中心之间的总和。众所周知,这个问题是APX硬,而目前的最佳近似率是6.357美元的初步接近率,其基础是针对该问题的标准LP[阿马迪安等人FOCS'17, SICOMP'20]。我们在本照会中表明,对艾哈迈迪安等人的分析稍稍作修改,导致略微改进了6.12903美元近似值。作为相关结果,我们还表明所提到的LP的一体化差距至少为$frac{16 ⁇ sqrt{5 ⁇ 15 ⁇ 2157美元。