Adaptive fully dynamic k-center with worst-case guarantees

Determine whether there exists a fully dynamic algorithm for the k-center clustering problem that is robust against an adaptive adversary and simultaneously guarantees (i) a constant-factor approximation, (ii) \~O(k) worst-case update time up to polylogarithmic factors, and (iii) O(1) worst-case recourse.

Background

Prior work (Bhattacharya, Costa, Farokhnejad, Lattanzi, and Parotsidis, ICML '25) achieved constant-factor approximation, near-linear-in-k expected amortized update time, and constant expected amortized recourse for fully dynamic k-center, but only against an oblivious adversary and with guarantees in expectation/amortized. Thus adaptivity and worst-case bounds were not settled.

The authors pose the central question of whether one can obtain, simultaneously, constant-factor approximation, ~O(k) worst-case update time (up to polylogarithmic factors), and O(1) worst-case recourse against an adaptive adversary. They subsequently claim to resolve this question affirmatively in their main results, but the question is explicitly stated as open immediately before presenting their contributions.

References

However, the fully dynamic algorithm in~\cref{thm:fully_dyn_almost_opt} assumes an oblivious adversary, and some of its guarantees hold only in expectation or in the amortized sense. For this reason, the following important question remains open: Is there a fully dynamic algorithm against an adaptive adversary that maintains a constant-factor approximate solution for the $k$-center clustering problem, with $\tilde{O}(k)$ worst-case update time and $O(1)$ worst-case recourse?

— Adaptive Fully Dynamic $k$-Center Clustering with (Near-)Optimal Worst-Case Guarantees  (2604.01726 - Grilnberger et al., 2 Apr 2026) in Introduction (Section 1), after Theorem 1