Dynamic Algorithms for Matroid Submodular Maximization
Submodular maximization under matroid and cardinality constraints are classical problems with a wide range of applications in machine learning, auction theory, and combinatorial optimization. In this paper, we consider these problems in the dynamic setting, where (1) we have oracle access to a monotone submodular function $f: 2^{V} \rightarrow \mathbb{R}^+$ and (2) we are given a sequence $\mathcal{S}$ of insertions and deletions of elements of an underlying ground set $V$. We develop the first fully dynamic $(4+\epsilon)$-approximation algorithm for the submodular maximization problem under the matroid constraint using an expected worst-case $O(k\log(k)\log^3{(k/\epsilon)})$ query complexity where $0 < \epsilon \le 1$. This resolves an open problem of Chen and Peng (STOC'22) and Lattanzi et al. (NeurIPS'20). As a byproduct, for the submodular maximization under the cardinality constraint $k$, we propose a parameterized (by the cardinality constraint $k$) dynamic algorithm that maintains a $(2+\epsilon)$-approximate solution of the sequence $\mathcal{S}$ at any time $t$ using an expected worst-case query complexity $O(k\epsilon^{-1}\log^2(k))$. This is the first dynamic algorithm for the problem that has a query complexity independent of the size of ground set $V$.
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