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A combinatorial algorithm for minimizing symmetric submodular functions
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The budgeted maximum coverage problem
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A combinatorial algorithm minimizing submodular functions in strongly polynomial time
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Combinatorial approximation algorithms for the maximum directed cut problem
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A combinatorial strongly polynomial algorithm for minimizing submodular functions
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Some optimal inapproximability results
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Improved Algorithms and Analysis for Secretary Problems and Generalizations
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Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT
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Adaptive limited-supply online auctions
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Optimal Inapproximability Results for Max-Cut and Other 2-Variable CSPs?
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A multiple-choice secretary algorithm with applications to online auctions
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On maximizing welfare when utility functions are subadditive
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On the complexity of approximating k-set packing
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Matroids, secretary problems, and online mechanisms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Maximizing Non-Monotone Submodular Functions
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Maximizing a Submodular Set Function Subject to a Matroid Constraint (Extended Abstract)
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A Knapsack Secretary Problem with Applications
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AAAI'07 Proceedings of the 22nd national conference on Artificial intelligence - Volume 1
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Secretary problems with competing employers
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A note on maximizing a submodular set function subject to a knapsack constraint
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Improved competitive ratios for submodular secretary problems
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Improved competitive ratio for the matroid secretary problem
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Secretary problems: laminar matroid and interval scheduling
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Matroid secretary problem in the random assignment model
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Interviewing secretaries in parallel
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Geometry of online packing linear programs
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Advances on matroid secretary problems: free order model and laminar case
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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 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 log2 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 Õ(√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(√n)-competitive algorithm for the subadditive case.