  Maximizing a monotone submodular function under various constraints is a
classical and intensively studied problem. However, in the single-pass
streaming model, where the elements arrive one by one and an algorithm can
store only a small fraction of input elements, there is much gap in our
knowledge, even though several approximation algorithms have been proposed in
the literature.
  In this work, we present the first lower bound on the approximation ratios
for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the
single-pass streaming model. Let $n$ be the number of elements in the stream.
Then, we prove that any (randomized) streaming algorithm for a cardinality
constraint with approximation ratio $\frac{2}{2+\sqrt{2}}+\varepsilon$ requires
$\Omega\left(\frac{n}{K^2}\right)$ space for any $\varepsilon>0$, where $K$ is
the size limit of the output set. We also prove that any (randomized) streaming
algorithm for a (partition) matroid constraint with approximation ratio
$\frac{K}{2K-1}+\varepsilon$ requires $\Omega\left(\frac{n}{K}\right)$ space
for any $\varepsilon>0$, where $K$ is the rank of the given matroid.
  In addition, we give streaming algorithms when we only have a weak oracle
with which we can only evaluate function values on feasible sets. Specifically,
we show weak-oracle streaming algorithms for cardinality and matroid
constraints with approximation ratios $\frac{K}{2K-1}$ and $\frac{1}{2}$,
respectively, whose space complexity is exponential in $K$ but is independent
of $n$. The former one exactly matches the known inapproximability result for a
cardinality constraint in the weak oracle model.
  The latter one almost matches our lower bound of $\frac{K}{2K-1}$ for a
matroid constraint, which almost settles the approximation ratio for a matroid
constraint that can be obtained by a streaming algorithm whose space complexity
is independent of $n$.
