Fully Dynamic $k$-Median with Near-Optimal Update Time and Recourse

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 $O(ε^{-1})$-approximation ratio with $\tilde{O}(k^ε)$ recourse and $\tilde{O}(k^{1+ε})$ 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.