  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.
