Adaptive Fully Dynamic $k$-Center Clustering with (Near-)Optimal Worst-Case Guarantees

Abstract

Given a sequence of adversarial point insertions and point deletions, is it possible to simultaneously optimize the approximation ratio, update time, and recourse for a $k$-clustering problem? If so, can this be achieved with worst-case guarantees against an adaptive adversary? These questions have garnered significant attention in recent years. Prior works by Bhattacharya, Costa, Garg, Lattanzi, and Parotsidis [FOCS '24] and by Bhattacharya, Costa, and Farokhnejad [STOC '25] have taken significant steps toward this direction for the $k$-median clustering problem and its generalization, the $(k, z)$-clustering problem. In this paper, we study the $k$-center clustering problem, which is one of the most classical and well-studied $k$-clustering problems. Recently, Bhattacharya, Costa, Farokhnejad, Lattanzi, and Parotsidis [ICML '25] provided an affirmative answer to the first question for the $k$-center clustering problem. However, their work did not resolve the second question, as their result provides only expected amortized guarantees against an oblivious adversary. In this work, we make significant progress and close the gap by answering both questions in the affirmative. Specifically, we show that the fully dynamic $k$-center clustering problem admits a constant-factor approximation, near-optimal worst-case update time, and constant worst-case recourse, even against an adaptive adversary. This is achieved by first developing a fully dynamic bicriteria approximation algorithm with (near-)optimal worst-case bounds, and then designing a suitable fully dynamic $k$-center algorithm with near-linear update time. For the fully dynamic bicriteria approximation algorithm, we establish the worst-case recourse and worst-case update time guarantees separately, and then merge them into a single algorithm through a simple yet elegant process.