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：考虑以下推广了最小半径和聚类的自然问题：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。目标是找到一个由$k$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$C\subseteq B(f_1,r_1)\cup\dots\cup B(f_k,r_k)$，并且对于某个递增函数$g:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq 0}$，$\sum_{i=1,\dots,k} g(r_i)$被最小化。此处$B(x,r)$表示以$x$为中心、半径为$r$的球。对于$(V,d)$是某个有界树宽的边加权图的最短路径度量的情形，我们提出了一种针对此类问题定制的动态规划算法，该算法实现了多项式运行时间，从而证明该问题相对于参数树宽属于$\mathsf{XP}$类。