Discretely Beyond 1/e: Guided Combinatorial Algorithms for Submodular Maximization

For constrained, not necessarily monotone submodular maximization, all known approximation algorithms with ratio greater than 1/e require continuous ideas, such as queries to the multilinear extension of a submodular function and its gradient, which are typically expensive to simulate with the original set function. For combinatorial algorithms, the best known approximation ratios for both size and matroid constraint are obtained by a simple randomized greedy algorithm of Buchbinder et al. [9]: 1/e ≈ 0.367 for size constraint and 0.281 for the matroid constraint in mathcal O (kn) queries, where k is the rank of the matroid. In this work, we develop the first combinatorial algorithms to break the 1/e barrier: we obtain approximation ratio of 0.385 in mathcal O (kn) queries to the submodular set function for size constraint, and 0.305 for a general matroid constraint. These are achieved by guiding the randomized greedy algorithm with a fast local search algorithm. Further, we develop deterministic versions of these algorithms, maintaining the same ratio and asymptotic time complexity. Finally, we develop a deterministic, nearly linear time algorithm with ratio 0.377.