In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to the closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on metrics. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art upper bound for maintaining $(1+\epsilon)$-approximate single-source distances in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

翻译：本文提出了首个在边更新的动态图上处理$k$-中心问题的有效算法。该问题的目标是通过选择$k$个中心将输入划分为$k$个集合，使得任意数据点到其最近中心的最大距离最小化。已知该问题在近似比优于2的情况下是NP难的。尽管在许多应用中输入可以自然地建模为图，但以往所有关于动态环境下$k$-中心问题的研究均基于度量空间。本文针对加权图提出了确定性递减$(2+\epsilon)$-近似算法和随机化递增$(4+\epsilon)$-近似算法，两者的均摊更新时间复杂度均为$kn^{o(1)}$。此外，我们通过一个归约得到全动态$(2+\epsilon)$-近似算法，其最坏情况更新时间复杂度与维护图中$(1+\epsilon)$-近似单源距离的最优上界相比，仅差一个因子$k$。实现该界是一个自然目标，因为每个顶点到其中心的近似距离可用于维护图直径的$(2+\epsilon)$-近似，而目前已知最快的直径近似算法同样依赖于维护近似单源距离。