Deterministic Algorithm and Faster Algorithm for Submodular Maximization subject to a Matroid Constraint

We study the problem of maximizing a monotone submodular function subject to a matroid constraint, and present for it a deterministic non-oblivious local search algorithm that has an approximation guarantee of $1 - 1/e - \varepsilon$ (for any $\varepsilon> 0$) and query complexity of $\tilde{O}_\varepsilon(nr)$, where $n$ is the size of the ground set and $r$ is the rank of the matroid. Our algorithm vastly improves over the previous state-of-the-art $0.5008$-approximation deterministic algorithm, and in fact, shows that there is no separation between the approximation guarantees that can be obtained by deterministic and randomized algorithms for the problem considered. The query complexity of our algorithm can be improved to $\tilde{O}_\varepsilon(n + r\sqrt{n})$ using randomization, which is nearly-linear for $r = O(\sqrt{n})$, and is always at least as good as the previous state-of-the-art algorithms.