New Algorithms and Hardness Results for Connected Clustering

2025-11-24

AI Summary

This paper studies the *k*-clustering problem under connectivity constraints: given an undirected connected graph *G* and an independent distance metric, partition its *n* vertices into *k* connected subgraphs (clusters) to optimize the *k*-center, min-sum-radii (MSR), or min-sum-diameter (MSD) objective. Methodologically, it integrates parameterized complexity analysis, dynamic programming over tree decompositions, and novel approximation techniques. Key contributions are threefold: (1) It establishes, assuming P ≠ NP, that the connected *k*-center problem admits no polynomial-time constant-factor approximation algorithm—resolving an open question; (2) For graphs of treewidth *tw*, it devises both an FPT exact algorithm and a constant-factor approximation algorithm; (3) On general graphs, it achieves a *(3 + ε)*-approximation for MSR and a *(4 + ε)*-approximation for MSD—improving the best-known MSD approximation ratio from *(6 + ε)* to *(4 + ε)*.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Clustering
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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 $Ω(n^{1- ε})$-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 $Ω(log^*(k))$-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 $(3 + ε)$ and connected MSD with an approximation factor of $(4 + ε)$. The latter also directly improves the best known approximation guarantee for unconstrained MSD from $(6 + ε)$ to $(4 + ε)$.

Problem

• Studying hardness and approximation for connected clustering problems with connectivity constraints
• Providing exact algorithms for bounded treewidth graphs and constant approximations
• Developing approximation algorithms for min-sum-radii and min-sum-diameter objectives

Innovation

• Exact algorithms for bounded treewidth graphs
• Constant approximation algorithms in FPT time
• Improved approximation factors for MSR and MSD