New Algorithms and Hardness Results for Connected Clustering

TL;DR

This paper introduces new algorithms and hardness results for connected clustering problems, showing significant complexity bounds, exact solutions for bounded treewidth graphs, and improved approximation algorithms for specific objectives.

Contribution

It establishes hardness results for connected k-center and k-median, provides polynomial-time algorithms for bounded treewidth graphs, and improves approximation factors for min-sum-radii and min-sum-diameter objectives.

Findings

Connected k-median is hard to approximate within (n^{1-psilon})

Polynomial-time algorithms for graphs with bounded treewidth

Improved approximation ratios for min-sum-radii and min-sum-diameter

Abstract

Connected clustering denotes a family of constrained clustering problems in which we are given a distance metric and an undirected connectivity graph $G$ that can be completely unrelated to the metric. The aim is to partition the $n$ vertices into a given number $k$ of clusters such that every cluster forms a connected subgraph of $G$ and a given clustering objective gets minimized. The constraint that the clusters are connected has applications in many different fields, like for example community detection and geodesy. So far, $k$-center and $k$-median have been studied in this setting. It has been shown that connected $k$-median is $\Omega(n^{1- \epsilon})$-hard to approximate which also carries over to the connected $k$-means problem, while for connected $k$-center it remained an open question whether one can find a constant approximation in polynomial time. We answer this question by providing an $\Omega(\log^*(k))$-hardness result for the problem. Given these hardness results, we study the problems on graphs with bounded treewidth. We provide exact algorithms that run in polynomial time if the treewidth $w$ is a constant. Furthermore, we obtain constant approximation algorithms that run in FPT time with respect to the parameter $\max(w,k)$. Additionally, we consider the min-sum-radii (MSR) and min-sum-diameter (MSD) objective. We prove that on general graphs connected MSR can be approximated with an approximation factor of $(3 + \epsilon)$ and connected MSD with an approximation factor of $(4 + \epsilon)$. The latter also directly improves the best known approximation guarantee for unconstrained MSD from $(6 + \epsilon)$ to $(4 + \epsilon)$.