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Submodular Secretary Problem and Extensions

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (RANDOM 2010, APPROX 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6302))

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(llog2 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 \(\tilde o(\sqrt{n})\)-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.

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Author information

Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08540, USA

    MohammadHossein Bateni

  2. AT&T Labs–Research, Florham Park, NJ, 07932, USA

    MohammadTaghi Hajiaghayi

  3. MIT, CSAIL, Cambridge, MA, 02139, USA

    Morteza Zadimoghaddam

Authors
  1. MohammadHossein Bateni
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  2. MohammadTaghi Hajiaghayi
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  3. Morteza Zadimoghaddam
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Editor information

Editors and Affiliations

  1. Universitat Politecnica de Catalunya, Departament de Llenguatges i Sistemes Informatics, Campus Nord - Ed Omega, Desp. 235, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Maria Serna

  2. Department of Computer Science, University of Haifa, Mount Carmel, 31905, Haifa, Israel

    Ronen Shaltiel

  3. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24098, Kiel, Germany

    Klaus Jansen

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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Bateni, M., Hajiaghayi, M., Zadimoghaddam, M. (2010). Submodular Secretary Problem and Extensions. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2010 2010. Lecture Notes in Computer Science, vol 6302. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15369-3_4

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  • DOI: https://doi.org/10.1007/978-3-642-15369-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15368-6

  • Online ISBN: 978-3-642-15369-3

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