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.

翻译：我们研究了具有$\textit{自适应}$边标签翻转的动态相关性聚类问题。在相关性聚类中，给定一个$n$顶点完全图，其边被标记为$(+)$或$(-)$，目标是最小化聚类间$(+)$边的总数与聚类内$(-)$边的数量。我们考虑具有对抗鲁棒性的动态设置，其中$\textit{自适应}$对抗者可以根据算法的当前输出翻转边的标签。我们的主要成果是一个随机化算法，该算法始终以$O(\log^{2}{n})$的摊还更新时间维护对最优相关性聚类的$O(1)$近似。在我们的工作之前，对于对抗鲁棒性设置，尚未有算法能在$O(1)$近似和$\text{polylog}{(n)}$更新时间下实现。我们进一步通过在合成和真实世界数据集上的实验验证了理论结果，并展示了具有竞争力的实证性能。我们的核心技术贡献是一个能以$\text{polylog}{(n)}$更新时间维护$\textit{稀疏-稠密分解}$的算法，这可能具有独立的研究价值。