Incremental (k, z)-Clustering on Graphs

• URL: http://arxiv.org/abs/2602.08542v1
• Date: Mon, 09 Feb 2026 11:43:10 GMT
• Title: Incremental (k, z)-Clustering on Graphs
• Authors: Emilio Cruciani, Sebastian Forster, Antonis Skarlatos,
• Abstract summary: We develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation.<n>In the first stage, we maintain a constant-factor bicriteria approximate solution of size $tildeO(k)$ with a total update time of $m1+o(1)$ over all adversarial edge insertions.<n>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.
• Score: 2.3322477552758234
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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