Submodular Secretary Problem with Shortlists

Abstract

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 $O(k \log k)$.