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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 O(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.

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We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves ( 1 /2 + ε)-approximation for the problem. This algorithm is the first deterministic algorithm known to improve over the 1 /2approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.