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 $(1/2-\varepsilon)$-approximation guarantee using $\tilde{O}(\varepsilon^{-1})$ 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 $\Omega(n)$ 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 $(1/2-\varepsilon)$-approximation guarantee for this problem while keeping the same adaptive and query complexities.Comment: Authors are listed in alphabetical orde

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 $(1/2-\varepsilon)$-approximation guarantee using $\tilde{O}(\varepsilon^{-1})$ 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 $\Omega(n)$ 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 $(1/2-\varepsilon)$-approximation guarantee for this problem while keeping the same adaptive and query complexities.Comment: Authors are listed in alphabetical orde

Nearly Linear-Time, Parallelizable Algorithms for Non-Monotone Submodular Maximization

We study parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a cardinality constraint $k$. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and query complexity, up to logarithmic factors in the size $n$ of the ground set, from $0.039 - \epsilon$ to $0.193 - \epsilon$. We provide two algorithms; the first has approximation ratio $1/6 - \epsilon$, adaptivity $O( \log n )$, and query complexity $O( n \log k )$, while the second has approximation ratio $0.193 - \epsilon$, adaptivity $O( \log^2 n )$, and query complexity $O(n \log k)$. Heuristic versions of our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with highly adaptive approximation algorithms.Comment: 24 pages, 2 figure