License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.62
URN: urn:nbn:de:0030-drops-187159
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18715/
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Hoefer, Martin ; Schewior, Kevin

Threshold Testing and Semi-Online Prophet Inequalities

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LIPIcs-ESA-2023-62.pdf (0.7 MB)

Abstract

We study threshold testing, an elementary probing model with the goal to choose a large value out of n i.i.d. random variables. An algorithm can test each variable X_i once for some threshold t_i, and the test returns binary feedback whether X_i ≥ t_i or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables.
Threshold testing is a semi-online variant of the gambler’s problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately 0.745 + o(1) as n → ∞. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least 0.869 - o(1). Moreover, we show that there are distributions that admit only a constant ratio c < 1, even when n → ∞. Finally, when each box can be tested multiple times (with n tests in total), we design an algorithm that achieves a ratio of 1 - o(1).

BibTeX - Entry

@InProceedings{hoefer_et_al:LIPIcs.ESA.2023.62,
  author =	{Hoefer, Martin and Schewior, Kevin},
  title =	{{Threshold Testing and Semi-Online Prophet Inequalities}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{62:1--62:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18715},
  URN =		{urn:nbn:de:0030-drops-187159},
  doi =		{10.4230/LIPIcs.ESA.2023.62},
  annote =	{Keywords: Prophet Inequalities, Testing, Stochastic Probing}
}

Keywords: Prophet Inequalities, Testing, Stochastic Probing
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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