We study the consistent k-center clustering problem. In this problem, the goal is to maintain a constant factor approximate $k$-center solution during a sequence of $n$ point insertions and deletions while minimizing the recourse, i.e., the number of changes made to the set of centers after each point insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that in the incremental setting, where deletions are not allowed, one can obtain $k \cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and $k$-median, and demonstrated a matching lower bound. However, no algorithm for the fully dynamic setting achieves less than the trivial $O(k)$ changes per update, which can be obtained by simply reclustering the full dataset after every update. In this work, we give the first algorithm for consistent $k$-center clustering for the fully dynamic setting, i.e., when both point insertions and deletions are allowed, and improves upon a trivial $O(k)$ recourse bound. Specifically, our algorithm maintains a constant factor approximate solution while ensuring worst-case constant recourse per update, which is optimal in the fully dynamic setting. Moreover, our algorithm is deterministic and is therefore correct even if an adaptive adversary chooses the insertions and deletions.

翻译：我们研究一致 $k$-中心聚类问题。在该问题中，目标是在包含 $n$ 个点的插入和删除序列中，维持一个常数因子近似的 $k$-中心解，同时最小化回溯开销（即每次点插入或删除后对中心集所做的更改次数）。Lattanzi 与 Vassilvitskii [ICML '12] 以及 Fichtenberger、Lattanzi、Norouzi-Fard 和 Svensson [SODA '21] 的前期工作表明，在不允许删除的增量设置中，对于 $k$-中心和 $k$-中位数问题，可实现 $k \cdot \textrm{polylog}(n) / n$ 的平摊回溯开销，并给出了匹配的下界。然而，在全动态设置中，尚无算法能实现低于每次更新 $O(k)$ 次更改的平凡界（该界可通过每次更新后对完整数据集重新聚类直接获得）。本文首次提出了全动态设置下（即允许点的插入和删除）的一致 $k$-中心聚类算法，并改进了平凡的 $O(k)$ 回溯界。具体而言，我们的算法在维持常数因子近似解的同时，确保每次更新的最坏情况回溯开销为常数，这在全动态设置中是最优的。此外，该算法是确定性的，因此即便自适应对手选择插入和删除操作，算法依然正确。