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

Abstract

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 $(f_1,r_1),\dots ,(f_k,r_k)\in F\times {\mathbb{R}}_{\ge 0}$ such that $C\subseteq B(f_1,r_1)\cup \cdots \cup B(f_k,r_k)$ and ${\sum }_{i=1,\dots ,k}g(r_i)$ is minimized for some increasing function $g:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}_{\ge 0}$. 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.

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