Maximizing a monotone submodular function under cardinality constraint $k$ is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS'20 by Lattanzi, Mitrovic, Norouzi{-}Fard, Tarnawski, Zadimoghaddam claims to obtain a dynamic algorithm for this problem with a $\frac{1}{2} -\epsilon$ approximation ratio and a query complexity bounded by $\mathrm{poly}(\log(n),\log(k),\epsilon^{-1})$. However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng at STOC'22 who show a matching lower bound for the problem -- any randomized algorithm with a $\frac{1}{2}+\epsilon$ approximation ratio must have an amortized query complexity that is polynomial in $n$. In this paper, we develop a simpler algorithm for the problem that maintains a $(\frac{1}{2}-\epsilon)$-approximate solution for submodular maximization under cardinality constraint $k$ using a polylogarithmic amortized update time.

翻译：在基数约束$k$下最大化单调子模函数是机器学习和数据库中的核心问题，具有许多基本应用，包括视频与数据摘要、推荐系统、特征提取、示例聚类和覆盖问题。我们研究该经典问题在全动态模型中的情况，其中给定底层元素集合上的插入和删除流，目标是通过快速更新时间维持近似解。近期一篇NeurIPS'20论文（Lattanzi, Mitrovic, Norouzi-Fard, Tarnawski, Zadimoghaddam）声称针对该问题提出了一种动态算法，具有$\frac{1}{2} -\epsilon$近似比，且查询复杂度上界为$\mathrm{poly}(\log(n),\log(k),\epsilon^{-1})$。然而，正如本文所述，该分析存在重要漏洞。考虑到Chen和Peng在STOC'22中的最新结果——他们证明任何具有$\frac{1}{2}+\epsilon$近似比的随机算法必须具有多项式于$n$的平摊查询复杂度——保持该问题具有多对数更新时间的动态算法显得尤为重要。本文针对该问题提出了一种更简单的算法，能够在基数约束$k$下维持子模最大化的$(\frac{1}{2}-\epsilon)$-近似解，且平摊更新时间为多对数级别。