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 Abbasi, Bhaskara, and Venkatasubramanian [2021] and Ghadiri, Samadi, and Vempala [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 $\Omega(\ell)$. In order to obtain our result, we introduce a strengthened LP relaxation and show that it has an integrality gap of $\Theta(\frac{\log \ell}{\log\log\ell})$ for a fixed $p$. Additionally, we present a bicriteria approximation algorithm, which generalizes the bicriteria approximation of Abbasi et al. [2021].

翻译：我们提出了一个 $( e @ O( p) /\\ log\ log\ ell} $( 折合) $( 折合), 美元( 折合) 美元( 折合), 美元( 折合) 美元( 折合), 美元( 折合), 美元( 折合), 美元( 折合), 美元( 折合), 美元( 折合) 。 更确切地说, 我们需要找到一套 美元( 折合), 美元( 折合), 美元( 折合), 美元( 折合) 美元( 折合) 。 我们的算法改进并概括了 美元( 折合) 美元( 折合) 。