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 parameterized (by the rank $k$ of a matroid $\mathcal{M}$) 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 $0 < \epsilon \le 1$. Chen and Peng at STOC'22 studied the complexity of this problem in the insertion-only dynamic model (a restricted version of the fully dynamic model where deletion is not allowed), and they raised the following important open question: *"for fully dynamic streams [sequences of insertions and deletions of elements], there is no known constant-factor approximation algorithm with poly(k) amortized queries for matroid constraints."* Our dynamic algorithm answers this question as well as an open problem of Lattanzi et al. (NeurIPS'20) affirmatively. 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 the expected amortized worst-case 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 $V$.

翻译：子模最大化在拟阵和基数约束下是经典问题，在机器学习、拍卖理论和组合优化中具有广泛应用。本文考虑动态设置下的这些问题，其中：(1) 我们可预言访问单调子模函数 $f: 2^{V} \rightarrow \mathbb{R}^+$；(2) 给定一个底层基集 $V$ 中元素的插入和删除序列 $\mathcal{S}$。我们针对拟阵约束下的子模最大化问题，提出了首个参数化（通过拟阵 $\mathcal{M}$ 的秩 $k$ ）动态 $(4+\epsilon)$-近似算法，其期望最坏情况下的查询复杂度为 $O(k\log(k)\log^3{(k/\epsilon)})$，其中 $0 < \epsilon \le 1$。Chen 和 Peng 在 STOC'22 上研究了该问题在仅插入动态模型（全动态模型的受限版本，不允许删除）中的复杂度，并提出了以下重要开放问题：*“对于全动态流[元素的插入和删除序列]，目前尚无针对拟阵约束的具有 poly(k) 平摊查询的常数因子近似算法。”* 我们的动态算法肯定地回答了该问题，同时也回应了 Lattanzi 等人（NeurIPS'20）的开放问题。作为副产品，针对基数约束 $k$ 下的子模最大化，我们提出了一种参数化（通过基数约束 $k$）动态算法，该算法在任何时刻 $t$ 维护序列 $\mathcal{S}$ 的 $(2+\epsilon)$-近似解，期望平摊最坏情况复杂度为 $O(k\epsilon^{-1}\log^2(k))$。这是该问题首个查询复杂度与基集 $V$ 大小无关的动态算法。