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DOI: 10.4230/LIPIcs.ISAAC.2018.65
URN: urn:nbn:de:0030-drops-100133
URL: http://drops.dagstuhl.de/opus/volltexte/2018/10013/
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Hoefer, Martin ; Wilhelmi, Lisa

Packing Returning Secretaries

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LIPIcs-ISAAC-2018-65.pdf (0.5 MB)


Abstract

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 accept 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 that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least 0.5721 - o(1) for growing n, 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 (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of Theta(n log n) is always sufficient. For matroids, we show that the expected number can be reduced to O(r log (n/r)), where r <=n/2 is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of O(r log n), where r is the size of the optimum matching. For general packing, we show a lower bound of Omega(n log log n), even when the size of the optimum is r = Theta(log n).

BibTeX - Entry

@InProceedings{hoefer_et_al:LIPIcs:2018:10013,
  author =	{Martin Hoefer and Lisa Wilhelmi},
  title =	{{Packing Returning Secretaries}},
  booktitle =	{29th International Symposium on Algorithms and Computation  (ISAAC 2018)},
  pages =	{65:1--65:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Wen-Lian Hsu and Der-Tsai Lee and Chung-Shou Liao},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10013},
  URN =		{urn:nbn:de:0030-drops-100133},
  doi =		{10.4230/LIPIcs.ISAAC.2018.65},
  annote =	{Keywords: Secretary Problem, Coupon Collector Problem, Matroids}
}

Keywords: Secretary Problem, Coupon Collector Problem, Matroids
Seminar: 29th International Symposium on Algorithms and Computation (ISAAC 2018)
Issue Date: 2018
Date of publication: 27.11.2018


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