Dynamic Algorithms for Matroid Submodular Maximization

TL;DR

This paper introduces the first fully dynamic algorithms with provable approximation guarantees for submodular maximization under matroid and cardinality constraints, significantly advancing the efficiency of dynamic combinatorial optimization.

Contribution

It presents the first fully dynamic $(4+\epsilon)$-approximation algorithm for matroid constraints and a parameterized $(2+\epsilon)$-approximate algorithm for cardinality constraints, with query complexities independent of ground set size.

Findings

First fully dynamic $(4+\epsilon)$-approximation for matroid constraints.

Dynamic $(2+\epsilon)$)-approximation for cardinality constraints with query complexity independent of ground set size.

Resolved open problems from STOC'22 and NeurIPS'20.

Abstract

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 $V$.