Fully Dynamic Adversarially Robust Correlation Clustering in Polylogarithmic Update Time

We study the dynamic correlation clustering problem with $\textit{adaptive}$ edge label flips. In correlation clustering, we are given a $n$-vertex complete graph whose edges are labeled either $(+)$ or $(-)$, and the goal is to minimize the total number of $(+)$ edges between clusters and the number of $(-)$ edges within clusters. We consider the dynamic setting with adversarial robustness, in which the $\textit{adaptive}$ adversary could flip the label of an edge based on the current output of the algorithm. Our main result is a randomized algorithm that always maintains an $O(1)$-approximation to the optimal correlation clustering with $O(\log^{2}{n})$ amortized update time. Prior to our work, no algorithm with $O(1)$-approximation and $\text{polylog}{(n)}$ update time for the adversarially robust setting was known. We further validate our theoretical results with experiments on synthetic and real-world datasets with competitive empirical performances. Our main technical ingredient is an algorithm that maintains $\textit{sparse-dense decomposition}$ with $\text{polylog}{(n)}$ update time, which could be of independent interest.

💡 Research Summary

Problem Setting
Correlation clustering seeks a partition of the vertices of a complete graph so that positive (+) edges lie inside clusters and negative (–) edges lie across clusters, minimizing the total number of disagreements. In many real‑world scenarios the edge labels evolve over time, prompting the study of dynamic algorithms that maintain a good clustering after each label flip. Prior dynamic work has focused on oblivious adversaries—those that must fix the entire update sequence in advance. In an adaptive (adversarially robust) setting, the adversary can observe the algorithm’s current output and choose the next edge flip to sabotage it. Existing pivot‑based dynamic algorithms break under such an adversary because their approximation guarantee relies on a uniformly random vertex permutation that can be learned and exploited.

Main Contribution
The paper presents the first dynamic algorithm that is both adversarially robust and polylogarithmic in update time while guaranteeing an O(1)‑approximation to the optimal correlation clustering cost. Specifically, the algorithm maintains an implicit clustering with amortized update time O(log² n) per edge label flip, regardless of how the adaptive adversary chooses the flips. This resolves an open problem in the literature.

Technical Core – Sparse‑Dense Decomposition
The algorithm builds on the sparse‑dense decomposition paradigm, which partitions a graph into two types of vertex groups:

1. Almost‑cliques – vertex sets that are nearly complete (few internal non‑edges) and have limited external connections.
2. Sparse vertices – vertices of low degree or those that are spread thinly across many almost‑cliques.

Earlier works (e.g., Ahn–Wang 2022, Charikar‑Liu‑Mossel 2021) showed that a static graph’s decomposition can be computed in Θ(n log² n) time and used to obtain a constant‑factor clustering. The novelty here is to maintain such a decomposition dynamically with only polylogarithmic effort.

Local Update Strategy
For each vertex u, the algorithm monitors the number of incident edge flips. When the count exceeds a small fraction (ε·deg(u)) of u’s current degree, the algorithm recomputes the sparse‑dense decomposition **only on the induced subgraph G