Designing efficient, effective, and consistent metric clustering algorithms is a significant challenge attracting growing attention. Traditional approaches focus on the stability of cluster centers; unfortunately, this neglects the real-world need for stable point labels, i.e., stable assignments of points to named sets (clusters). In this paper, we address this gap by initiating the study of label-consistent metric clustering. We first introduce a new notion of consistency, measuring the label distance between two consecutive solutions. Then, armed with this new definition, we design new consistent approximation algorithms for the classical $k$-center and $k$-median problems.

翻译：设计高效、有效且具有一致性的度量聚类算法是一项重大挑战，正日益受到关注。传统方法聚焦于聚类中心的稳定性；然而，这忽视了实际应用中对稳定点标签的需求，即点与命名集合（聚类）之间稳定分配的需求。本文通过开创性地研究标签一致性度量聚类来弥补这一空白。我们首先提出一种新的“一致性”概念，用于度量两个连续解之间的标签距离。随后，基于这一新定义，我们为经典的 $k$-center 与 $k$-median 问题设计了新的具有一致性的近似算法。