Approximation Algorithms for Socially Fair Clustering
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We present an (e^O(p)logℓ/loglogℓ)-approximation algorithm for socially fair clustering with the ℓ_p-objective. In this problem, we are given a set of points in a metric space. Each point belongs to one (or several) of ℓ groups. The goal is to find a k-medians, k-means, or, more generally, ℓ_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 ∑_u in group j d(u,C)^p. The socially fair clustering problem was independently proposed by Abbasi, Bhaskara, and Venkatasubramanian [2021] and Ghadiri, Samadi, and Vempala [2021]. Our algorithm improves and generalizes their O(ℓ)-approximation algorithms for the problem. The natural LP relaxation for the problem has an integrality gap of Ω(ℓ). In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of Θ(logℓ/loglogℓ) for a fixed p. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].
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