Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact $(k, z)$-clustering solution in the induced shortest-path metric. While efficient dynamic $k$-center approximation algorithms on graphs exist [Cruciani et al. SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic $(k,z)$-clustering problem. As the main result of this paper, we develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}λ} m)$, where $λ\geq 1$ is an arbitrary fixed constant. Our incremental algorithm consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size $\tilde{O}(k)$ with a total update time of $m^{1+o(1)}$ over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant, a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate $(k,z)$-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static $(k,z)$-clustering algorithm.

翻译：给定一个加权无向图、聚类数量 $k$ 以及指数 $z$，图上的 $(k, z)$-聚类问题的目标是选择 $k$ 个顶点作为中心，以最小化每个顶点到其最近中心的距离的 $z$ 次幂之和。在动态设定中，图受到对抗性边更新的影响，目标是在诱导的最短路径度量中显式地维护一个精确的 $(k, z)$-聚类解。尽管存在高效的动态图 $k$-中心近似算法 [Cruciani et al. SODA 2024]，但据我们所知，尚无先前工作为动态 $(k,z)$-聚类问题提供类似结果。作为本文的主要成果，我们开发了一种随机增量式 $(k, z)$-聚类算法，该算法以高概率在经历边插入的图中维持一个常数因子近似，其总更新时间为 $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}λ} m)$，其中 $λ\geq 1$ 是一个任意固定常数。我们的增量式算法包含两个阶段。在第一阶段，我们维护一个大小为 $\tilde{O}(k)$ 的常数因子双准则近似解，其在整个对抗性边插入过程中的总更新时间为 $m^{1+o(1)}$。这个第一阶段是对 Mettu 和 Plaxton [Machine Learning 2004] 的双准则近似算法在增量图上的复杂改编。我们的一个关键技术成果是，在他们的算法中，半径可以被假设为是非递减的，同时近似比保持为常数，这一特性可能具有独立的研究价值。在第二阶段，我们在由双准则近似解诱导出的动态加权实例上，维护一个常数因子近似的 $(k,z)$-聚类解。对于这个子问题，我们采用了动态生成稀疏子图算法结合静态 $(k,z)$-聚类算法。