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

Bogdan Adrian Manghiuc, He Sun

Keywords: Hierarchical clustering, graph algorithms, spectral methods

TL;DR: We present $O(1)$-approximation algorithms for hierarchical clustering when the input graph exhibits a clears structure of clusters.

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.

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Code: https://github.com/bgmang/hierarchical-clustering-well-clustered-graphs.git

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2112.09055/code)