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].

中文翻译：

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社会公平聚类的近似算法

我们提出了一个$（e ^ {O（p）} \ frac {\ log \ ell} {\ log \ log \ ell}）$近似算法，用于具有$ \ ell_p $目标的社会公平聚类。在这个问题上，我们在度量空间中获得了一组点。每个点都属于$ \ ell $组中的一个（或几个）。目标是找到同时对所有组都有好处的$ k $中位数，$ k $-均值或更普遍的$ \ ell_p $聚类。更确切地说，我们需要找到一组$ k $中心$ C $，以最小化所有组\ j_的$ j $的最大值，其中$ \ sum_ {u \ text {在}}} d（u，C）^ p $。社会公平集群问题是由阿巴西，巴斯卡拉和文卡塔苏布拉曼尼亚人[2021]以及加迪里，萨马迪和范帕拉[2021]独立提出的。我们的算法对问题进行了改进并推广了他们的$ O（\ ell）$逼近算法。该问题的自然LP松弛具有$ \ Omega（\ ell）$的完整性缺口。为了获得我们的结果，我们引入了增强的LP松弛，并证明了对于固定的$ p $，它具有$ \ Theta（\ frac {\ log \ ell} {\ log \ log \ ell}）$的完整性缺口。此外，我们提出了一种双标准近似算法，该算法概括了Abbasi等人的双标准近似。[2021]。