Approximation Algorithms for Socially Fair Clustering
This addresses fairness in clustering algorithms for multiple demographic groups, offering a significant theoretical improvement over prior work.
The paper tackles the problem of socially fair clustering by developing an approximation algorithm with a ratio of e^{O(p)} * (log ℓ / log log ℓ), improving upon previous O(ℓ)-approximation methods, and introduces a strengthened LP relaxation with an integrality gap of Θ(log ℓ / log log ℓ).
We present an $(e^{O(p)} \frac{\log \ell}{\log\log\ell})$-approximation algorithm for socially fair clustering with the $\ell_p$-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of $\ell$ groups. The goal is to find a $k$-medians, $k$-means, or, more generally, $\ell_p$-clustering that is simultaneously good for all of the groups. More precisely, we need to find a set of $k$ centers $C$ so as to minimize the maximum over all groups $j$ of $\sum_{u \text{ in group }j} d(u,C)^p$. The socially fair clustering problem was independently proposed by Ghadiri, Samadi, and Vempala [2021] and Abbasi, Bhaskara, and Venkatasubramanian [2021]. Our algorithm improves and generalizes their $O(\ell)$-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of $Ω(\ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $Θ(\frac{\log \ell}{\log\log\ell})$ for a fixed $p$. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].