Linear-Time Algorithms for Adaptive Submodular Maximization
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In this paper, we develop fast algorithms for two stochastic submodular maximization problems. We start with the well-studied adaptive submodular maximization problem subject to a cardinality constraint. We develop the first linear-time algorithm which achieves a (1-1/e-ϵ) approximation ratio. Notably, the time complexity of our algorithm is O(nlog1/ϵ) (number of function evaluations) which is independent of the cardinality constraint, where n is the size of the ground set. Then we introduce the concept of fully adaptive submodularity, and develop a linear-time algorithm for maximizing a fully adaptive submoudular function subject to a partition matroid constraint. We show that our algorithm achieves a 1-1/e-ϵ/4-2/e-2ϵ approximation ratio using only O(nlog1/ϵ) number of function evaluations.
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