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International Symposium on Algorithms and Computation

ISAAC 2015: Algorithms and Computation pp 129–139Cite as

The Secretary Problem with a Choice Function

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9472))

Abstract

In the classical secretary problem, a decision-maker is willing to hire the best secretary out of n applicants that arrive in a random order, and the goal is to maximize the probability of choosing the best applicant. In this paper, we introduce the secretary problem with a choice function. The choice function represents the preference of the decision-maker. In this problem, the decision-maker hires some applicants, and the goal is to maximize the probability of choosing the best set of applicants defined by the choice function. We see that the secretary problem with a path-independent choice function generalizes secretary version of the stable matching problem, the maximum weight bipartite matching problem, and the maximum weight base problem in a matroid. When the choice function is path-independent, we provide an algorithm that succeeds with probability at least \(1/e^k\) where k is the maximum size of the choice, and prove that this is the best possible. Moreover, for the non-path-independent case, we prove that the success probability goes to arbitrary small for any algorithm even if the maximum size of the choice is 2.

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Acknowledgement

The author thanks Tomomi Matsui and Keisuke Bando for valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Number 26887014.

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Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Yasushi Kawase

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  1. Yasushi Kawase
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Correspondence to Yasushi Kawase .

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Editors and Affiliations

  1. Masdar Institute, Abu Dhabi, United Arab Emirates

    Khaled Elbassioni

  2. Kyoto University, Kyoto, Japan

    Kazuhisa Makino

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Kawase, Y. (2015). The Secretary Problem with a Choice Function. In: Elbassioni, K., Makino, K. (eds) Algorithms and Computation. ISAAC 2015. Lecture Notes in Computer Science(), vol 9472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48971-0_12

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  • DOI: https://doi.org/10.1007/978-3-662-48971-0_12

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-48970-3

  • Online ISBN: 978-3-662-48971-0

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