In metric $k$-clustering, we are given as input a set of $n$ points in a general metric space, and we have to pick $k$ centers and cluster the input points around these chosen centers, so as to minimize an appropriate objective function. In recent years, significant effort has been devoted to the study of metric $k$-clustering problems in a dynamic setting, where the input keeps changing via updates (point insertions/deletions), and we have to maintain a good clustering throughout these updates. The performance of such a dynamic algorithm is measured in terms of three parameters: (i) Approximation ratio, which signifies the quality of the maintained solution, (ii) Recourse, which signifies how stable the maintained solution is, and (iii) Update time, which signifies the efficiency of the algorithm. We consider the metric $k$-median problem, where the objective is the sum of the distances of the points to their nearest centers. We design the first dynamic algorithm for this problem with near-optimal guarantees across all three performance measures (up to a constant factor in approximation ratio, and polylogarithmic factors in recourse and update time). Specifically, we obtain a $O(1)$-approximation algorithm for dynamic metric $k$-median with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Prior to our work, the state-of-the-art here was the recent result of [Bhattacharya et al., FOCS'24], who obtained $O(\epsilon^{-1})$-approximation ratio with $\tilde{O}(k^{\epsilon})$ recourse and $\tilde{O}(k^{1+\epsilon})$ update time. We achieve our results by carefully synthesizing the concept of robust centers introduced in [Fichtenberger et al., SODA'21] along with the randomized local search subroutine from [Bhattacharya et al., FOCS'24], in addition to several key technical insights of our own.

翻译：在度量空间k-聚类问题中，我们给定一个一般度量空间中包含n个点的输入集，需要选取k个中心并将输入点围绕这些选定的中心进行聚类，以最小化适当的目标函数。近年来，研究者们致力于研究动态场景下的度量k-聚类问题，其中输入通过更新（点的插入/删除）不断变化，我们需要在整个更新过程中维持良好的聚类效果。此类动态算法的性能通过三个参数衡量：（i）近似比，反映所维持解的质量；（ii）调整代价，反映所维持解的稳定性；（iii）更新时间，反映算法的效率。本文研究度量k-中值问题，其目标函数为各点到其最近中心距离的总和。我们为该问题设计了首个在三个性能指标上均具有接近最优保证的动态算法（近似比为常数因子，调整代价和更新时间为多对数因子）。具体而言，我们得到了一个具有$\tilde{O}(1)$调整代价和$\tilde{O}(k)$更新时间的$O(1)$近似动态度量k-中值算法。在本研究之前，该领域的最先进成果是[Bhattacharya等人，FOCS'24]近期提出的算法，其获得了$O(\epsilon^{-1})$近似比、$\tilde{O}(k^{\epsilon})$调整代价和$\tilde{O}(k^{1+\epsilon})$更新时间。我们通过精心整合[Fichtenberger等人，SODA'21]提出的鲁棒中心概念与[Bhattacharya等人，FOCS'24]的随机局部搜索子程序，并结合若干关键的技术创新，实现了上述结果。