Relative Error Fair Clustering in the Weak-Strong Oracle Model

TL;DR

This paper introduces a novel approach for fair clustering that balances the use of costly accurate distance measurements and cheaper inaccurate ones, achieving near-optimal solutions with minimal strong oracle queries.

Contribution

It presents the first $(1+ ext{varepsilon})$-coresets for fair $k$-median clustering using polylogarithmic strong oracle queries, advancing fair clustering with inaccurate data.

Findings

Achieved $(1+ ext{varepsilon})$-coresets for fair $k$-median clustering.

Extended results to standard and $(k,z)$-clustering without fairness constraints.

Reduced the number of strong oracle queries needed for near-optimal clustering.

Abstract

We study fair clustering problems in a setting where distance information is obtained from two sources: a strong oracle providing exact distances, but at a high cost, and a weak oracle providing potentially inaccurate distance estimates at a low cost. The goal is to produce a near-optimal fair clustering on $n$ input points with a minimum number of strong oracle queries. This models the increasingly common trade-off between accurate but expensive similarity measures (e.g., large-scale embeddings) and cheaper but inaccurate alternatives. The study of fair clustering in the model is motivated by the important quest of achieving fairness with the presence of inaccurate information. We achieve the first $(1+\varepsilon)$-coresets for fair $k$-median clustering using $\text{poly}\left(\frac{k}{\varepsilon}\cdot\log n\right)$ queries to the strong oracle. Furthermore, our results imply coresets for the standard setting (without fairness constraints), and we could in fact obtain $(1+\varepsilon)$-coresets for $(k,z)$-clustering for general $z=O(1)$ with a similar number of strong oracle queries. In contrast, previous results achieved a constant-factor $(>10)$ approximation for the standard $k$-clustering problems, and no previous work considered the fair $k$-median clustering problem.