Many real-world applications pose challenges in incorporating fairness constraints into the $k$-center clustering problem, where the dataset consists of $m$ demographic groups, each with a specified upper bound on the number of centers to ensure fairness. Focusing on big data scenarios, this paper addresses the problem in a streaming setting, where data points arrive one by one sequentially in a continuous stream. Leveraging a structure called the $λ$-independent center set, we propose a one-pass streaming algorithm that first computes a reserved set of points during the streaming process. Then, for the post-streaming process, we propose an approach for selecting centers from the reserved point set by analyzing all three possible cases, transforming the most complicated one into a specially constrained vertex cover problem in an auxiliary graph. Our algorithm achieves a tight approximation ratio of 5 while consuming $O(k\log n)$ memory. It can also be readily adapted to solve the offline fair $k$-center problem, achieving a 3-approximation ratio that matches the current state of the art. Furthermore, we extend our approach to a semi-structured data stream, where data points from each group arrive in batches. In this setting, we present a 3-approximation algorithm for $m = 2$ and a 4-approximation algorithm for general $m$. Lastly, we conduct extensive experiments to evaluate the performance of our approaches, demonstrating that they outperform existing baselines in both clustering cost and runtime efficiency.

翻译：许多实际应用在将公平性约束融入$k$-中心聚类问题时面临挑战，其中数据集包含$m$个人口统计组，每组均设有确保公平性的中心数量上限。针对大数据场景，本文在流式设置下处理该问题，即数据点以连续流的形式逐个顺序到达。借助一种称为$λ$-独立中心集的结构，我们提出了一种单遍流式算法，该算法首先在流式处理过程中计算一个保留点集。随后，针对流后处理，我们提出了一种从保留点集中选择中心的方法，通过分析所有三种可能情况，将最复杂的情况转化为辅助图中特殊约束的顶点覆盖问题。我们的算法实现了紧致的5倍近似比，同时仅消耗$O(k\log n)$内存。该算法也可直接适用于求解离线公平$k$-中心问题，获得与当前最优水平匹配的3倍近似比。此外，我们将方法扩展到半结构化数据流场景，其中每组数据点以批次形式到达。在此设置下，我们针对$m=2$提出了3倍近似算法，针对一般$m$提出了4倍近似算法。最后，我们进行了大量实验以评估所提方法的性能，结果表明其在聚类成本和运行效率方面均优于现有基线方法。