In the Euclidean $k$-means problems we are given as input a set of $n$ points in $\mathbb{R}^d$ and the goal is to find a set of $k$ points $C\subseteq \mathbb{R}^d$, so as to minimize the sum of the squared Euclidean distances from each point in $P$ to its closest center in $C$. In this paper, we formally explore connections between the $k$-coloring problem on graphs and the Euclidean $k$-means problem. Our results are as follows: $\bullet$ For all $k\ge 3$, we provide a simple reduction from the $k$-coloring problem on regular graphs to the Euclidean $k$-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean $2$-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. $\bullet$ In the other direction, we mimic the $O(1.7297^n)$ time algorithm of Williams [TCS'05] for the max-cut of problem on $n$ vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in $2^n\cdot \text{poly}(n,d)$ time. $\bullet$ We prove similar results and connections as above for the Euclidean $k$-min-sum problem.

翻译：在欧几里得$k$-均值问题中，我们被给予一组$\mathbb{R}^d$空间中的$n$个点作为输入，目标是找到一组$k$个点$C\subseteq \mathbb{R}^d$，以最小化从$P$中每个点到其最近中心$C$的欧几里得距离平方和。本文正式探讨了图上$k$-着色问题与欧几里得$k$-均值问题之间的关联。我们的研究结果如下：$\bullet$ 对于所有$k\ge 3$，我们给出了从正则图上的$k$-着色问题到欧几里得$k$-均值问题的简单归约。此外，我们的技术可扩展至从结构化最大割问题（可视为部分2-着色问题）到欧几里得$2$-均值问题的归约。因此，我们为欧几里得2-均值问题的NP难性提供了一个简洁的替代证明。$\bullet$ 在相反方向上，我们模拟Williams [TCS'05]针对$n$个顶点上最大割问题的$O(1.7297^n)$时间算法，以得到具有相同运行时间的欧几里得2-均值问题算法，改进了在$2^n\cdot \text{poly}(n,d)$时间内运行的朴素穷举搜索。$\bullet$ 我们针对欧几里得$k$-最小和问题证明了与上述类似的结果和关联。