Abstract: We study combinatorial, 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 of the ground set, from 0.039 to nearly 0.193. 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 state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.