Article

Submodular secretary problem and extensions

Authors:

MohammadHossein Bateni,

MohammadTaghi Hajiaghayi,

Morteza ZadimoghaddamAuthors Info & Claims

APPROX/RANDOM'10: Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques

Pages 39 - 52

Published: 01 September 2010 Publication History

Abstract

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.

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cover image Guide Proceedings

APPROX/RANDOM'10: Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques

September 2010

781 pages

ISBN:3642153682

Editors:
Maria Serna
Universitat Politècnica de Catalunya, Dept. de Llenguatges i Sistemes Informàtics, Barcelona, Spain
,
Ronen Shaltiel
University of Haifa, Department of Computer Science, Haifa, Israel
,
Klaus Jansen
University of Kiel, Department of Computer Science, Kiel, Germany
,
José Rolim
University of Geneva, Centre Universitaire d'Informatique, Carouge, Switzerland

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Berlin, Heidelberg

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Published: 01 September 2010

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Ehsani SHajiaghayi MKesselheim TSingla SCzumaj A(2018)Prophet secretary for combinatorial auctions and matroidsProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3174304.3175316(700-714)Online publication date: 7-Jan-2018
https://dl.acm.org/doi/10.5555/3174304.3175316
Babaioff MImmorlica NKempe DKleinberg R(2018)Matroid Secretary ProblemsJournal of the ACM10.1145/321251265:6(1-26)Online publication date: 19-Nov-2018
https://dl.acm.org/doi/10.1145/3212512
Epasto ALattanzi SVassilvitskii SZadimoghaddam MBarrett RCummings RAgichtein EGabrilovich E(2017)Submodular Optimization Over Sliding WindowsProceedings of the 26th International Conference on World Wide Web10.1145/3038912.3052699(421-430)Online publication date: 3-Apr-2017
https://dl.acm.org/doi/10.1145/3038912.3052699
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