Practical Parallel Algorithms for Submodular Maximization Subject to a Knapsack Constraint with Nearly Optimal Adaptivity.

Submodular maximization has wide applications in machine learning and data mining, where massive datasets have brought the great need for designing efficient and parallelizable algorithms. One measure of the parallelizability of a sub-modular maximization algorithm is its adaptivity complexity, which indicates the number of sequential rounds where a polynomial number of queries to the objective function can be executed in parallel. In this paper, we study the problem of non-monotone submodular maximization subject to a knapsack constraint, and propose the first combinatorial algorithm achieving an (8 + ŝ )-approximation under O (log n ) adaptive complexity, which is optimal up to a factor of O (log log n ). Moreover, under slightly larger adaptivity, we also propose approximation algorithms with nearly optimal query complexity of O(n) , while achieving better approximation ratios. We show that our algorithms can also be applied to the special case of submodular maximization subject to a cardinality constraint, and achieve performance bounds comparable with those of state-of-the-art algorithms. Finally, the effectiveness of our approach is demonstrated by extensive experiments on real-world applications.