Incremental (k, z)-Clustering on Graphs

TL;DR

This work develops a randomized incremental framework for the graph $(k,z)$-clustering problem under adversarial edge updates. It first builds a constant-factor bicriteria approximation of size $O(k)$ and then reduces the problem to a dynamic, distance-preserving sparse instance on a center set via a dynamic spanner, solving a static $(k,z)$-clustering on this reduced instance. The primary theoretical contribution is a total update time of $ ilde{O}(k m^{1+o(1)}+ k^{1+1/\lambda} m)$ with high probability, and an amortized near-optimal time of $ ilde{O}(k n^{o(1)}+ k^{1+1/\lambda})$, for fixed $\lambda\ge 1$, while maintaining an $O(1)$-approximation to the optimal $(k,z)$-clustering. The combination of a carefully structured MP-bi variant, a leaking-set mechanism, and a dynamic spanner-based reduction yields practical, scalable dynamic clustering on graphs in the incremental setting, with potential impact on near-linear-time graph clustering in evolving networks.

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.

Figures (1)

• Figure 1: The blue thick regions depict the $i$-th approximate ball $B_i$ for a level $i \in [0, t]$, and the larger blue vertices represent the $i$-th candidate set $S_i$. The black lines indicate distances. In the right figure, the brown lines and dots depict new shorter distances of vertices that enter $B_i$ due to the adversarial edge insertion. Left figure: The $i$-th approximate ball before the adversarial edge insertion. Right figure: After the adversarial edge insertion, some vertices enter the $i$-th approximate ball $B_i$ (brown vertices). Since the $i$-th radius $\nu_i$ is decreased (the dashed blue region indicates the old radius), some vertices enter the temporary leaking set $Z$.

Theorems & Definitions (119)

• theorem 1.1
• theorem 1.2
• definition 3.1: $(k,z)$-clustering problem on graphs
• definition 3.2: $\rho$-approximate $(k,z)$-clustering problem
• definition 3.3: $(\alpha, \beta)$-bicriteria approximation
• definition 3.4: $(\alpha, \beta)$-bicriteria approximate assignment
• definition 3.4: weighted $(k,z)$-clustering problem on graphs
• lemma 3.5: incremental $(1+\epsilon)$-approximate SSSP liu2025incremental
• lemma 3.6: dubhashi2009concentration
• proof