The $k$-center problem is a central optimization problem with numerous applications for machine learning, data mining, and communication networks. Despite extensive study in various scenarios, it surprisingly has not been thoroughly explored in the traditional distributed setting, where the communication graph of a network also defines the distance metric. We initiate the study of the $k$-center problem in a setting where the underlying metric is the graph's shortest path metric in three canonical distributed settings: the LOCAL, CONGEST, and CLIQUE models. Our results encompass constant-factor approximation algorithms and lower bounds in these models, as well as hardness results for the bi-criteria approximation setting.

翻译：$k$-中心问题是机器学习和数据挖掘领域的一个核心优化问题，在通信网络等场景中具有广泛应用。尽管该问题已在多种情境下得到广泛研究，但令人惊讶的是，在传统分布式场景中——其中网络的通信图同时定义了距离度量——该问题尚未得到深入探索。我们首次在底层度量采用图最短路径度量的三种经典分布式模型（LOCAL、CONGEST和CLIQUE）中系统研究$k$-中心问题。我们的研究成果包括这些模型中的常数因子近似算法与下界证明，以及对双标准近似场景的硬度结果分析。