Submodular Maximization with Matroid and Packing Constraints in Parallel

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a 1-1/e-ε approximation for monotone functions and a 1/e-ε approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is O(log2n/ε3), which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a 1/e-ε approximation using O(log(n/ε) log(1/ε) log(n+m)/ ε2) parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a 1-1/e-ε approximation in O(log(n/ε)log(m)/ε2) parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.