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 usingÕ(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.
REF considered the unconstrained submodular maximization problem and proposed the first algorithm that achieves the optimal approximation guarantee in a constant number of adaptive rounds.
Unconstrained submodular maximization with constant adaptive complexity
STOC 2019
Mathematics Computer Science