DASH: A Distributed and Parallelizable Algorithm for Size-Constrained Submodular Maximization

MapReduce (MR) algorithms for maximizing monotone, submodular functions subject to a cardinality constraint (SMCC) are currently restricted to the use of the linearadaptive (non-parallelizable) algorithm GREEDY. Lowadaptive algorithms do not satisfy the requirements of these distributed MR frameworks, thereby limiting their performance. We study the SMCC problem in a distributed setting and propose the first MR algorithms with sublinear adaptive complexity. Our algorithms, R-DASH, T-DASH and G-DASH provide 0.316 -ε, 3/8 -ε, and 1 -1/e -ε approximation ratios, respectively, with nearly optimal adaptive complexity and nearly linear time complexity. Additionally, we provide a framework to increase, under some mild assumptions, the maximum permissible cardinality constraint from O(n/ℓ 2 ) of prior MR algorithms to O(n/ℓ), where n is the data size and ℓ is the number of machines; under a stronger condition on the objective function, we increase the maximum constraint value to n. Finally, we provide empirical evidence to demonstrate that our sublinear-adaptive, distributed algorithms provide orders of magnitude faster runtime compared to current state-of-the-art distributed algorithms.