Unconstrained submodular maximization with constant adaptive complexity

In this paper, we consider the unconstrained submodular maximization problem. We propose the first algorithm for this problem that achieves a tight (1/2 − ε)-approximation guarantee using Õ(ε⁻¹) adaptive rounds and a linear number of function evaluations. No previously known algorithm for this problem achieves an approximation ratio better than 1/3 using less than Ω(n) rounds of adaptivity, where n is the size of the ground set. Moreover, our algorithm easily extends to the maximization of a non-negative continuous DR-submodular function subject to a box constraint, and achieves a tight (1/2 − ε)-approximation guarantee for this problem while keeping the same adaptive and query complexities.