Given points from an arbitrary metric space and a sequence of point updates sent by an adversary, what is the minimum recourse per update (i.e., the minimum number of changes needed to the set of centers after an update), in order to maintain a constant-factor approximation to a $k$-clustering problem? This question has received attention in recent years under the name consistent clustering. Previous works by Lattanzi and Vassilvitskii [ICLM '17] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] studied $k$-clustering objectives, including the $k$-center and the $k$-median objectives, under only point insertions. In this paper we study the $k$-center objective in the fully dynamic setting, where the update is either a point insertion or a point deletion. Before our work, {\L}\k{a}cki, Haeupler, Grunau, Rozho\v{n}, and Jayaram [SODA '24] gave a deterministic fully dynamic constant-factor approximation algorithm for the $k$-center objective with worst-case recourse of $2$ per update. In this work, we prove that the $k$-center clustering problem admits optimal recourse bounds by developing a deterministic fully dynamic constant-factor approximation algorithm with worst-case recourse of $1$ per update. Moreover our algorithm performs simple choices based on light data structures, and thus is arguably more direct and faster than the previous one which uses a sophisticated combinatorial structure. Additionally, we develop a new deterministic decremental algorithm and a new deterministic incremental algorithm, both of which maintain a $6$-approximate $k$-center solution with worst-case recourse of $1$ per update. Our incremental algorithm improves over the $8$-approximation algorithm by Charikar, Chekuri, Feder, and Motwani [STOC '97]. Finally, we remark that since all three of our algorithms are deterministic, they work against an adaptive adversary.

翻译：给定来自任意度量空间的点集以及由敌手发送的点更新序列，为了在每次更新后维持对$k$-聚类问题的常数因子近似，每次更新所需的最小调整代价（即更新后对中心集合的最小改动次数）是多少？近年来，这一问题以“一致聚类”的名称受到关注。Lattanzi与Vassilvitskii [ICML '17] 以及Fichtenberger、Lattanzi、Norouzi-Fard和Svensson [SODA '21] 的前期工作仅针对点插入情形研究了$k$-聚类目标，包括$k$-中心和$k$-中位数目标。本文我们在完全动态设置下研究$k$-中心目标，其中更新操作可以是点插入或点删除。在我们工作之前，{\L}\k{a}cki、Haeupler、Grunau、Rozho\v{n}和Jayaram [SODA '24] 提出了一种确定性完全动态常数因子近似算法用于$k$-中心目标，其最坏情况每次更新调整代价为$2$。本工作中，我们证明$k$-中心聚类问题可实现最优调整代价界，通过开发一种确定性完全动态常数因子近似算法，其最坏情况每次更新调整代价为$1$。此外，我们的算法基于轻量数据结构执行简单选择，因此相比先前使用复杂组合结构的算法更直接且更快速。另外，我们开发了一种新的确定性递减算法和一种新的确定性递增算法，两者均能以每次更新最坏情况调整代价$1$维持$6$-近似的$k$-中心解。我们的递增算法改进了Charikar、Chekuri、Feder和Motwani [STOC '97] 的$8$-近似算法。最后我们指出，由于我们的三种算法均为确定性算法，它们可应对自适应敌手。