The fairness of clustering algorithms has gained widespread attention across various areas, including machine learning, In this paper, we study fair $k$-means clustering in Euclidean space. Given a dataset comprising several groups, the fairness constraint requires that each cluster should contain a proportion of points from each group within specified lower and upper bounds. Due to these fairness constraints, determining the optimal locations of $k$ centers is a quite challenging task. We propose a novel ``Relax and Merge'' framework that returns a $(1+4\rho + O(\epsilon))$-approximate solution, where $\rho$ is the approximate ratio of an off-the-shelf vanilla $k$-means algorithm and $O(\epsilon)$ can be an arbitrarily small positive number. If equipped with a PTAS of $k$-means, our solution can achieve an approximation ratio of $(5+O(\epsilon))$ with only a slight violation of the fairness constraints, which improves the current state-of-the-art approximation guarantee. Furthermore, using our framework, we can also obtain a $(1+4\rho +O(\epsilon))$-approximate solution for the $k$-sparse Wasserstein Barycenter problem, which is a fundamental optimization problem in the field of optimal transport, and a $(2+6\rho)$-approximate solution for the strictly fair $k$-means clustering with no violation, both of which are better than the current state-of-the-art methods. In addition, the empirical results demonstrate that our proposed algorithm can significantly outperform baseline approaches in terms of clustering cost.

翻译：聚类算法的公平性已在包括机器学习在内的多个领域引起广泛关注。本文研究欧氏空间中的公平$k$-均值聚类问题。给定包含若干组别的数据集，公平性约束要求每个聚类中来自各组的点所占比例需在指定的上下界范围内。由于这些公平性约束的存在，确定$k$个中心的最优位置成为一项极具挑战性的任务。我们提出一种新颖的“松弛与合并”框架，该框架可返回$(1+4\rho + O(\epsilon))$近似解，其中$\rho$为现有标准$k$-均值算法的近似比，$O(\epsilon)$可为任意小的正数。若配备$k$-均值的PTAS（多项式时间近似方案），我们的解在仅轻微违反公平性约束的情况下即可达到$(5+O(\epsilon))$的近似比，这改进了当前最先进的近似保证。此外，利用本框架，我们还能为最优传输领域的基础优化问题——$k$-稀疏Wasserstein质心问题获得$(1+4\rho +O(\epsilon))$近似解，并为严格无违反的公平$k$-均值聚类获得$(2+6\rho)$近似解，这两项结果均优于当前最先进的方法。实验结果表明，我们提出的算法在聚类代价方面显著优于基线方法。