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A Bicriterion Concentration Inequality and Prophet Inequalities for k-Fold Matroid Unions

Authors Noga Alon , Nick Gravin , Tristan Pollner, Aviad Rubinstein , Hongao Wang , S. Matthew Weinberg , Qianfan Zhang

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ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Prophet Inequalities
  • Online Contention Resolution Schemes
  • Concentration Inequalities

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Abstract

We investigate prophet inequalities with competitive ratios approaching 1, seeking to generalize k-uniform matroids. We first show that large girth does not suffice: for all k, there exists a matroid of girth ≥ k and a prophet inequality instance on that matroid whose optimal competitive ratio is 1/2. Next, we show k-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio 1-O(√{(log k)/k}) for any k-fold matroid union. Our prophet inequality follows from an online contention resolution scheme.
The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone 1-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a 1-Lipschitz function that is not (approximately) self-bounding.

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Noga Alon, Nick Gravin, Tristan Pollner, Aviad Rubinstein, Hongao Wang, S. Matthew Weinberg, and Qianfan Zhang. A Bicriterion Concentration Inequality and Prophet Inequalities for k-Fold Matroid Unions. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.4

Author Details

Noga Alon
  • Department of Mathematics, Princeton University, NJ, USA
  • Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv, Israel
Nick Gravin
  • Key Laboratory of Interdisciplinary Research of Computation and Economics, Shanghai University of Finance and Economics, China
Tristan Pollner
  • Department of Management Science and Engineering, Stanford University, CA, USA
Aviad Rubinstein
  • Department of Computer Science, Stanford University, CA, USA
Hongao Wang
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
S. Matthew Weinberg
  • Department of Computer Science, Princeton University, NJ, USA
Qianfan Zhang
  • Department of Computer Science, Princeton University, NJ, USA

Funding

  • Alon, Noga: Research supported in part by NSF grant DMS-2154082.
  • Gravin, Nick: Research is supported by National Key R & D Program of China (2023YFA1009500), NSFC grant 61932002, "the Fundamental Research Funds for the Central Universities in China".
  • Rubinstein, Aviad: Research supported in part by NSF CCF-1954927, and a David and Lucile Packard Fellowship.
  • Weinberg, S. Matthew: Research supported in part by NSF CCF-1955205. During Professor Weinberg’s development of this paper, he participated as an expert witness on behalf of the State of Texas in ongoing litigation against Google (the "Google Litigation").
  • Zhang, Qianfan: Research supported in part by NSF CCF-1955205.

Acknowledgements

The authors are grateful to the anonymous reviewers for helpful feedback on the initial submission of this work.

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