Dynamic Algorithms for Matroid Submodular Maximization
This addresses incremental improvements in dynamic algorithms for combinatorial optimization problems relevant to machine learning and auction theory.
The paper tackles dynamic submodular maximization under matroid and cardinality constraints by developing the first fully dynamic (4+ε)-approximation algorithm with expected worst-case O(k log(k) log^3(k/ε)) query complexity, resolving an open problem, and also proposes a dynamic (2+ε)-approximation algorithm for cardinality constraints with query complexity independent of ground set size.
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 $(4+ε)$-approximation algorithm for the submodular maximization problem under the matroid constraint using an expected worst-case $O(k\log(k)\log^3{(k/ε)})$ query complexity where $0 < ε\le 1$. 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 $(2+ε)$-approximate solution of the sequence $\mathcal{S}$ at any time $t$ using an expected worst-case query complexity $O(kε^{-1}\log^2(k))$. This is the first dynamic algorithm for the problem that has a query complexity independent of the size of ground set $V$.