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

TL;DR

This paper presents a fully dynamic algorithm for the $k$-center clustering problem that achieves near-optimal worst-case guarantees, including approximation ratio, update time, and recourse, even against adaptive adversaries.

Contribution

It introduces the first fully dynamic $k$-center clustering algorithm with constant-factor approximation, near-linear update time, and constant recourse against adaptive adversaries.

Findings

Achieves constant-factor approximation in fully dynamic setting.

Provides near-linear worst-case update time.

Ensures constant worst-case recourse.

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.