Submodular secretary problems with extensions
Online auction is the 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 article, 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 (i.e., a knapsack constraint assigns a nonnegative cost to each secretary, and requires that the total cost of all the secretaries employed be no more than a budget value) 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.