We consider the k-diameter clustering problem, where the goal is to partition a set of points in a metric space into $k$ clusters, minimizing the maximum distance between any two points in the same cluster. In general metrics, k-diameter is known to be NP-hard, while it has a $2$-approximation algorithm (Gonzalez'85). Complementing this algorithm, it is known that k-diameter is NP-hard to approximate within a factor better than $2$ in the $\ell_1$ and $\ell_\infty$ metrics, and within a factor of $1.969$ in the $\ell_2$ metric (Feder-Greene'88). When $k\geq 3$ is fixed, k-diameter remains NP-hard to approximate within a factor better than $2$ in the $\ell_\infty$ metric (Megiddo'90). However, its approximability in this setting has not previously been studied in the $\ell_1$ and $\ell_2$ metrics, though a $1.415$-approximation algorithm in the $\ell_2$ metric follows from a known result (Badoiu et al.'02). In this paper, we address the remaining gap by showing new hardness of approximation results that hold even when $k=3$. Specifically, we prove that 3-diameter is NP-hard to approximate within a factor better than $1.5$ in the $\ell_1$ metric, and within a factor of $1.304$ in the $\ell_2$ metric.

翻译：我们考虑k-直径聚类问题，其目标是将度量空间中的一组点划分为$k$个簇，最小化同一簇中任意两点之间的最大距离。在一般度量空间中，k-直径已知是NP难的，但存在一个$2$-近似算法（Gonzalez'85）。作为对该算法的补充，已知在$\ell_1$和$\ell_\infty$度量中，k-直径在优于$2$的因子内近似是NP难的，而在$\ell_2$度量中，该因子为$1.969$（Feder-Greene'88）。当$k\geq 3$固定时，k-直径在$\ell_\infty$度量中仍然在优于$2$的因子内近似是NP难的（Megiddo'90）。然而，在$\ell_1$和$\ell_2$度量中，该场景下的可近似性此前尚未被研究，尽管在$\ell_2$度量中存在一个$1.415$-近似算法来自已知结果（Badoiu et al.'02）。在本文中，我们通过展示新的硬度近似结果来填补这一空白，这些结果即使在$k=3$时也成立。具体而言，我们证明在$\ell_1$度量中，3-直径在优于$1.5$的因子内近似是NP难的，而在$\ell_2$度量中，该因子为$1.304$。