Threshold Testing and Semi-Online Prophet Inequalities

Authors Martin Hoefer , Kevin Schewior

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Author Details

Martin Hoefer
  • Institute of Computer Science, Goethe University Frankfurt, Germany
Kevin Schewior
  • Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark


The authors gratefully acknowledge discussions with Daniel Schmand and Luca von der Brelie.

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Martin Hoefer and Kevin Schewior. Threshold Testing and Semi-Online Prophet Inequalities. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Online algorithms
  • Theory of computation → Markov decision processes
  • Prophet Inequalities
  • Testing
  • Stochastic Probing


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