Hierarchical $\mathcal{F}$-Clustering: Approximation and Hardness of Clustering into Trees and Bounded Diameter Graphs

2026-07-14

AI Summary

This work addresses the hierarchical $\mathcal{F}$-clustering problem, where the stopping condition is relaxed from singleton clusters to clusters belonging to a specified graph class—such as trees or graphs of bounded diameter. The authors propose a unified linear programming–based approximation framework that models the underlying flat clustering as an integer linear program and employs tailored rounding strategies to construct a clustering tree. This framework yields the first $\mathcal{O}(\log n \cdot \log \log n)$-approximation algorithm for tree clustering and an $\mathcal{O}(\log n)$-approximation for bounded-diameter graph clustering, while also characterizing structural conditions on graph classes under which the approach applies. Furthermore, under the Small Set Expansion hypothesis, the paper establishes hardness results showing that constant-factor approximations are infeasible, thereby delineating tight theoretical bounds.

Abstract

Consider the following variation on the Hierarchical Clustering problem: Usually, while building a hierarchical clustering, one recursively partitions the data until each cluster becomes a singleton. We relax the halting condition of the recursive process to stop whenever the remaining cluster is a graph belonging to a class $\mathcal{F}$. We call this problem Hierarchical $\mathcal{F}$-Clustering and we measure the quality of any solution using adapted Dasgupta's clustering objective. We study two natural choices of $\mathcal{F}$: trees and graphs of bounded diameter. We present the first polynomial time $\mathcal{O}(\log n\cdot\log\log n)$ and $\mathcal{O}(\log n)$-approximation algorithms for clustering into trees and bounded diameter graphs respectively. Our main technical contribution is a framework for approximating such problems based on linear programming. In fact, we characterize graphs classes $\mathcal{F}$ for which our approach can be applied and show that it includes both trees and bounded diameter graphs. However, our ideas are not limited to them and might be useful for other structures as well. Broadly speaking, our framework applies whenever the corresponding flat clustering problem, which we call $p_{\mathcal{F}}$-Partitioning, admits a natural ILP formulation together with a rounding procedure with provable approximation guarantees. Intuitively, given a set of vertices called terminals, the problem is to find an edge set whose removal results in satisfying certain vertex-dependent structural predicate for each terminal. We then use these ingredients to build clustering trees with the aforementioned approximation guarantees. To complement these results, we show that both Hierarchical Clustering into trees and into bounded diameter graphs cannot be approximated within any constant factor under the Small Set Expansion Hypothesis.

Problem

• Hierarchical Clustering
• Tree Clustering
• Bounded Diameter Graphs
• Approximation Algorithms
• Clustering Objective

Innovation

• Hierarchical ℱ-Clustering
• approximation algorithms
• linear programming framework
• bounded diameter graphs
• tree clustering