  In submodular $k$-secretary problem, the goal is to select $k$ items in a
randomly ordered input so as to maximize the expected value of a given monotone
submodular function on the set of selected items. In this paper, we introduce a
relaxation of this problem, which we refer to as submodular $k$-secretary
problem with shortlists. In the proposed problem setting, the algorithm is
allowed to choose more than $k$ items as part of a shortlist. Then, after
seeing the entire input, the algorithm can choose a subset of size $k$ from the
bigger set of items in the shortlist. We are interested in understanding to
what extent this relaxation can improve the achievable competitive ratio for
the submodular $k$-secretary problem. In particular, using an $O(k)$ shortlist,
can an online algorithm achieve a competitive ratio close to the best
achievable online approximation factor for this problem? We answer this
question affirmatively by giving a polynomial time algorithm that achieves a
$1-1/e-\epsilon -O(k^{-1})$ competitive ratio for any constant $\epsilon > 0$,
using a shortlist of size $\eta_\epsilon(k) = O(k)$. Also, for the special case
of m-submodular functions, we demonstrate an algorithm that achieves a
$1-\epsilon$ competitive ratio for any constant $\epsilon > 0$, using an $O(1)$
shortlist. Finally, we show that our algorithm can be implemented in the
streaming setting using a memory buffer of size $\eta_\epsilon(k) = O(k)$ to
achieve a $1 - 1/e - \epsilon-O(k^{-1})$ approximation for submodular function
maximization in the random order streaming model. This substantially improves
upon the previously best known approximation factor of $1/2 + 8 \times
10^{-14}$ [Norouzi-Fard et al. 2018] that used a memory buffer of size $O(k
\log k)$.
