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Clustering is a fundamental building block of modern statistical analysispipelines. Fair clustering has seen much attention from the machine learningcommunity in recent years. We are some of the first to study fairness in thecontext of hierarchical clustering, after the results of Ahmadian et al. fromNeurIPS in 2020. We evaluate our results using Dasgupta's cost function,perhaps one of the most prevalent theoretical metrics for hierarchicalclustering evaluation. Our work vastly improves the previous$O(n^{5/6}poly\log(n))$ fair approximation for cost to a near polylogarithmic$O(n^\delta poly\log(n))$ fair approximation for any constant $\delta\in(0,1)$.This result establishes a cost-fairness tradeoff and extends to broaderfairness constraints than the previous work. We also show how to alter existinghierarchical clusterings to guarantee fairness and cluster balance across anylevel in the hierarchy.

Generalized Reductions: Making any Hierarchical Clustering Fair and Balanced with Low Cost