Packing returning secretaries

  • We study online secretary problems with returns in combinatorial packing domains with n candidates that arrive sequentially over time in random order. The goal is to determine a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All 2n arrivals occur in random order. We propose a simple 0.5‐competitive algorithm. For the online bipartite matching problem, we obtain an algorithm with ratio at least 0.5721 − o(1), and an algorithm with ratio at least 0.5459 for all n ≥ 1. We extend all algorithms and ratios to k ≥ 2 arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed. We focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Θ(n log n) is always sufficient. For bipartite matching, we can show a tight bound of O(r log n), where r is the size of the optimum matching. For matroids, we can improve this further to a tight bound of O(r′ log(n/r′)), where r′ is the minimum rank of the matroid and the dual matroid.

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Author:Martin HoeferORCiDGND, Lisa Wilhelmi
Parent Title (German):Networks
Place of publication:New York, NY
Document Type:Article
Date of Publication (online):2020/11/16
Date of first Publication:2020/11/26
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2021/02/18
Tag:coupon collector problem; matching; matroids, online algorithm; packing problem; secretary problem
Page Number:18
First Page:1
Last Page:18
Institutes:Informatik und Mathematik / Informatik
Dewey Decimal Classification:0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik
Licence (German):License LogoCreative Commons - Namensnennung 4.0