  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
$1-1/e-\epsilon$ approximation for monotone functions and a $1/e-\epsilon$
approximation for non-monotone functions, which nearly matches the best
guarantees known in the fully adaptive setting. The number of rounds of
adaptivity is $O(\log^2{n}/\epsilon^3)$, 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
$1/e-\epsilon$ approximation using $O(\log(n/\epsilon) \log(1/\epsilon)
\log(n+m)/ \epsilon^2)$ parallel rounds, which is again an exponential speedup
in parallel time over the existing algorithms. For monotone functions, we
obtain a $1-1/e-\epsilon$ approximation in
$O(\log(n/\epsilon)\log(m)/\epsilon^2)$ 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.
