We consider the following natural problem that generalizes min-sum-radii clustering: Given is $k\in\mathbb{N}$ as well as some metric space $(V,d)$ where $V=F\cup C$ for facilities $F$ and clients $C$. The goal is to find a clustering given by $k$ facility-radius pairs $(f_1,r_1),\dots,(f_k,r_k)\in F\times\mathbb{R}_{\geq 0}$ such that $C\subseteq B(f_1,r_1)\cup\dots\cup B(f_k,r_k)$ and $\sum_{i=1,\dots,k} g(r_i)$ is minimized for some increasing function $g:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$. Here, $B(x,r)$ is the radius-$r$ ball centered at $x$. For the case that $(V,d)$ is the shortest-path metric of some edge-weighted graph of bounded treewidth, we present a dynamic program that is tailored to this class of problems and achieves a polynomial running time, establishing that the problem is in $\mathsf{XP}$ with parameter treewidth.

翻译：我们考虑以下推广最小半径和聚类问题的自然问题：给定 $k\in\mathbb{N}$ 以及度量空间 $(V,d)$，其中 $V=F\cup C$，$F$ 表示设施，$C$ 表示客户。目标是寻找由 $k$ 个设施-半径对 $(f_1,r_1),\dots,(f_k,r_k)\in F\times\mathbb{R}_{\geq 0}$ 构成的聚类，使得 $C\subseteq B(f_1,r_1)\cup\dots\cup B(f_k,r_k)$，且对于某个递增函数 $g:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$，$\sum_{i=1,\dots,k} g(r_i)$ 最小化。其中 $B(x,r)$ 表示以 $x$ 为中心、半径为 $r$ 的球。当 $(V,d)$ 为某个边赋权且有界树宽图的最短路径度量时，我们提出了一种针对此类问题的动态规划算法，该算法具有多项式运行时间，从而证明该问题属于参数树宽的 $\mathsf{XP}$ 类。