Data summarization tasks are often modeled as $k$-clustering problems, where the goal is to choose $k$ data points, called cluster centers, that best represent the dataset by minimizing a clustering objective. A popular objective is to minimize the maximum distance between any data point and its nearest center, which is formalized as the $k$-center problem. While in some applications all data points can be chosen as centers, in the general setting, centers must be chosen from a predefined subset of points, referred as facilities or suppliers; this is known as the $k$-supplier problem. In this work, we focus on fair data summarization modeled as the fair $k$-supplier problem, where data consists of several groups, and a minimum number of centers must be selected from each group while minimizing the $k$-supplier objective. The groups can be disjoint or overlapping, leading to two distinct problem variants each with different computational complexity. We present $3$-approximation algorithms for both variants, improving the previously known factor of $5$. For disjoint groups, our algorithm runs in polynomial time, while for overlapping groups, we present a fixed-parameter tractable algorithm, where the exponential runtime depends only on the number of groups and centers. We show that these approximation factors match the theoretical lower bounds, assuming standard complexity theory conjectures. Finally, using an open-source implementation, we demonstrate the scalability of our algorithms on large synthetic datasets and assess the price of fairness on real-world data, comparing solution quality with and without fairness constraints.

本文解决了数据摘要任务中的公正聚类问题，尤其是公平 k-供应商问题，该问题要求在各组中选择一定数量的中心点以最小化 k-供应商目标。我们提出了对两个不同变体的 3-近似算法，研究表明，这些算法在大规模数据集上具有良好的可扩展性，并且相对于没有公平约束的解决方案，其效果具有重要的实用性和影响。

公正聚类用于数据摘要：改进的近似算法及复杂性洞察