Clustering Graphs of Bounded Treewidth to Minimize the Sum of Radius-Dependent Costs

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 $(f1,r1),\dots,(fk,rk)\in F\times\mathbb{R}{\geq 0}$ such that $C\subseteq B(f1,r1)\cup\dots\cup B(fk,rk)$ and $\sum{i=1,\dots,k} g(ri)$ 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.