The $k$-center problem is one of the best-studied and most intuitive clustering formulations. It asks, given a set of $n$ points in a metric space, for $k$ of the points to be designated as cluster centers, so that the maximum distance of an input point to its nearest center is minimized. Gonzalez's greedy algorithm from 1985 is a simple and efficient way to find a $2$-approximate solution. The algorithm has the attractive feature of \emph{incrementality}: it outputs the centers one by one, with a guaranteed $2$-approximation for every prefix of the obtained sequence of centers. Incrementality imposes a geometric constraint on how solutions can be built, and it is natural to ask whether this comes at a price in the quality of the solution. It is known that in polynomial time, the approximation ratio of $2$ is best possible, assuming $P \neq NP$. In this paper we show that even with \emph{unlimited} computational power, the factor $2$ cannot be improved, if the solution is required to be built incrementally. The lower bound construction imposes a tradeoff between all $n$ levels of the clustering simultaneously; it was obtained with the help of ChatGPT, an aspect we discuss in Section 3 of the paper.

翻译：$k$-中心问题是最经典且最直观的聚类形式之一。该问题要求：给定度量空间中的$n$个点，选取其中$k$个点作为聚类中心，使所有输入点到其最近中心的最大距离最小化。Gonzalez于1985年提出的贪心算法是求解该问题的一种简单高效的方法，能够获得$2$-近似解。该算法具有卓越的增量特性：它逐个输出中心点，且所得中心序列的每个前缀都能保证$2$-近似比。增量性对解的构建施加了几何约束，自然会引发一个疑问：这种特性是否以牺牲解的质量为代价？已知在多项式时间内，若假设$P \neq NP$，则$2$为最优近似比。本文证明，即使拥有无限计算能力，若要求解必须增量式构建，因子$2$同样无法改进。该下界构造同时对聚类的全部$n$个层级施加权衡约束；我们借助ChatGPT（详见论文第3节讨论）完成了这一构造。