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−ϵ−O(k−1) competitive ratio for any constant ϵ>0,
using a shortlist of size ηϵ(k)=O(k). Also, for the special case
of m-submodular functions, we demonstrate an algorithm that achieves a
1−ϵ competitive ratio for any constant ϵ>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 ηϵ(k)=O(k) to
achieve a 1−1/e−ϵ−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+8×10−14 [Norouzi-Fard et al. 2018] that used a memory buffer of size O(klogk)