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算法，该算法始终以$O(\log^{2}{n})$的摊还更新时间维持对最优相关性聚类的$O(1)$近似。在我们的工作之前，对于对抗鲁棒设置，尚不存在具有$O(1)$近似和$\text{polylog}{(n)}$更新时间的算法。我们进一步在合成和真实世界数据集上通过实验验证了我们的理论结果，并展示了具有竞争力的实证性能。我们的主要技术贡献是一个能以$\text{polylog}{(n)}$更新时间维持$\textit{稀疏-稠密分解}$的算法，这可能具有独立的研究价值。