Optimal Algorithm and Lower Bound for Submodular Maximization

In this work, we present the first linear query complexity algorithm for maximizing a monotone submodular function subject to a cardinality constraint, which achieves the approximation ratio of $(1-1/e-\varepsilon)$ using $O(n/\varepsilon)$ queries. To the best of our knowledge, this is the first deterministic algorithm that achieves the almost optimal approximation and optimal query complexity simultaneously. Query complexity lower bound of submodular maximization problems is also studied in this paper. We show that there exists no (randomized) $(1/4+\varepsilon)$-approximate algorithm using $o(n/\log n)$ queries for unconstrained submodular maximization. Combining with existing results, we present a complete characterization of the query complexity of unconstrained submodular maximization.