Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the $k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius problem, a near-quadratic time algorithm with expected running time $O(n^2 \log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n \log^2 \log n)$ time. We generalize this result to any constant dimension $d$, for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time algorithm. Additionally, we give a near-quadratic time algorithm for $3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log n)$ time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution.

翻译：给定欧几里得平面上的 $n$ 个点集，$k$-MinSumRadius 问题要求使用 $k$ 个圆盘覆盖该点集，目标是最小化所有圆盘半径之和。经过一系列相关问题的长期研究，最终发现该问题存在多项式时间算法 [GKKPV~'12]；然而，该算法的运行时间为 $O(n^{881})$，因此其意义主要停留在理论层面。$k$-MinSumRadius 问题的一个实际且结构上有趣的特殊情形是 $k$ 较小的情况。对于 $2$-MinSumRadius 问题，30 多年前已给出期望运行时间为 $O(n^2 \log^2 n \log^2 \log n)$ 的近二次时间算法 [Eppstein~'92]。我们提出了这一结果的首个改进，即一种期望运行时间为 $O(n \log^2 n \log^2 \log n)$ 的近线性时间算法，用于计算 $2$-MinSumRadius。我们将这一结果推广到任意常数维度 $d$，并给出了运行时间为 $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ 的算法。此外，我们针对平面上的 $3$-MinSumRadius 问题，提出了一种期望运行时间为 $O(n^2 \log^2 n \log^2 \log n)$ 的近二次时间算法。所有这些算法均基于对最优解中一个惊人简单结构的揭示：我们可以指定线性数量的直线，每条直线将其中一个簇与最优解中的其余簇分隔开。