Hierarchical Clustering: $O(1)$-Approximation for Well-Clustered Graphs

TL;DR

This paper introduces two polynomial-time algorithms that achieve constant-factor approximations for hierarchical clustering on well-structured graphs, improving upon previous methods especially for graphs with clear cluster formations.

Contribution

The paper presents the first $O(1)$-approximation algorithms for hierarchical clustering on well-clustered graphs, extending beyond stochastic models and simplifying previous recursive approaches.

Findings

Algorithms outperform previous methods on synthetic data.

Algorithms effectively handle graphs with well-defined clusters.

Empirical results demonstrate practical efficiency and accuracy.

Abstract

Hierarchical clustering studies a recursive partition of a data set into clusters of successively smaller size, and is a fundamental problem in data analysis. In this work we study the cost function for hierarchical clustering introduced by Dasgupta, and present two polynomial-time approximation algorithms: Our first result is an $O(1)$-approximation algorithm for graphs of high conductance. Our simple construction bypasses complicated recursive routines of finding sparse cuts known in the literature. Our second and main result is an $O(1)$-approximation algorithm for a wide family of graphs that exhibit a well-defined structure of clusters. This result generalises the previous state-of-the-art, which holds only for graphs generated from stochastic models. The significance of our work is demonstrated by the empirical analysis on both synthetic and real-world data sets, on which our presented algorithm outperforms the previously proposed algorithm for graphs with a well-defined cluster structure.