# Submodular Secretary Problem and Extensions

Online auction is an essence of many modern markets, particularly networked markets, in which information about goods, agents, and outcomes is revealed over a period of time, and the agents must make irrevocable decisions without knowing future information. Optimal stopping theory, especially the classic "secretary problem", is a powerful tool for analyzing such online scenarios which generally require optimizing an objective function over the input. The secretary problem and its generalization the "multiple-choice secretary problem" were under a thorough study in the literature. In this paper, we consider a very general setting of the latter problem called the "submodular secretary problem", in which the goal is to select k secretaries so as to maximize the expectation of a (not necessarily monotone) submodular function which defines efficiency of the selected secretarial group based on their overlapping skills. We present the first constant-competitive algorithm for this case. In a more general setting in which selected secretaries should form an independent (feasible) set in each of l given matroids as well, we obtain an O(l log^2 r)-competitive algorithm generalizing several previous results, where r is the maximum rank of the matroids. Another generalization is to consider l knapsack constraints instead of the matroid constraints, for which we present an O(l)-competitive algorithm. In a sharp contrast, we show for a more general setting of "subadditive secretary problem, there is no o~(sqrt(n))-competitive algorithm and thus submodular functions are the most general functions to consider for constant competitiveness in our setting. We complement this result by giving a matching O(sqrt(n))-competitive algorithm for the subadditive case. At the end, we consider some special cases of our general setting as well.