Submodular Secretary Problem with Shortlists

Shipra Agrawal, Mohammad Shadravan, Cliff Stein

Published 2018-09-13Version 1

In \nameOfProblem, 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 \nameOfProblemSL. 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 \nameOfProblem. In particular, using an O(k) shortlist, can an online algorithm achieve a competitive ratio close to the best achievable offline 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). This is especially surprising considering that the best known competitive ratio (in polynomial time) for the \nameOfProblem~is (1/e-O(k^{-1/2}))(1-1/e) \cite{kesselheim}. Further, for the special case of m-submodular functions, we demonstrate an algorithm that achieves 1-epsilon competitive ratio for any constant epsilon>0, using an O(1) shortlist. For submodular function maximization under random order streaming model and k-cardinality constraint, 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. This substantially improves upon \cite{norouzi}, which achieved the previously best known approximation factor of 1/2 + 8\times 10^{-14} using O(k\log k) memory.