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

TL;DR

This paper introduces a dynamic programming approach for clustering graphs with bounded treewidth to minimize radius-dependent costs, extending min-sum-radii clustering to a broader class of graph metrics.

Contribution

It develops a polynomial-time dynamic programming algorithm for a generalized clustering problem on graphs with bounded treewidth, showing the problem is in XP with respect to treewidth.

Findings

Algorithm runs in polynomial time for bounded treewidth graphs.

The problem is classified as XP parameterized by treewidth.

Extends min-sum-radii clustering to shortest-path metrics of bounded treewidth graphs.

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 $(f_1,r_1),\dots,(f_k,r_k)\in F\times\mathbb{R}_{\geq 0}$ 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.