Nearly Linear-Time, Parallelizable Algorithms for Non-Monotone Submodular Maximization
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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 - ϵ to 0.193 - ϵ. We provide two algorithms; the first has approximation ratio 1/6 - ϵ, adaptivity O( log n ), and query complexity O( n log k ), while the second has approximation ratio 0.193 - ϵ, 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.
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