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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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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.

Cite As Get BibTex

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

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.

Subject Classification

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

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