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Sample-Based Matroid Prophet Inequalities

Hu Fu, Pinyan Lu, Zhihao Gavin Tang, Hongxun Wu, Jinzhao Wu, Qianfan Zhang

TL;DR

The paper addresses matroid prophet inequalities when distributions are unknown and accessible only via samples. It introduces a quantile-based reduction to OCRSs and designs a sample-efficient, adaptive matroid OCRS that tolerates dependence introduced by sampling, achieving a $(\frac{1}{4}-\varepsilon)$-competitive guarantee with $O_{\varepsilon}(\log^4 n)$ samples. The core ideas combine medians of exchange thresholds, a weighted strong exchange lemma, and a randomized chain-decomposition OCRS to overcome adaptivity. This advances constant-factor results for general matroids under limited information and connects to the matroid secretary problem via robust reductions.

Abstract

We study matroid prophet inequalities when distributions are unknown and accessible only through samples. While single-sample prophet inequalities for special matroids are known, no constant-factor competitive algorithm with even a sublinear number of samples was known for general matroids. Adding more to the stake, the single-sample version of the question for general matroids has close (two-way) connections with the long-standing matroid secretary conjecture. In this work, we give a $(\frac14 - \varepsilon)$-competitive matroid prophet inequality with only $O_\varepsilon(\mathrm{poly} \log n)$ samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with $O_\varepsilon(\log n)$ samples, and (ii) a $(\frac14 - \varepsilon)$-selectable matroid OCRS with $O_\varepsilon(\mathrm{poly} \log n)$ samples which carefully addresses an adaptivity challenge.

Sample-Based Matroid Prophet Inequalities

TL;DR

The paper addresses matroid prophet inequalities when distributions are unknown and accessible only via samples. It introduces a quantile-based reduction to OCRSs and designs a sample-efficient, adaptive matroid OCRS that tolerates dependence introduced by sampling, achieving a -competitive guarantee with samples. The core ideas combine medians of exchange thresholds, a weighted strong exchange lemma, and a randomized chain-decomposition OCRS to overcome adaptivity. This advances constant-factor results for general matroids under limited information and connects to the matroid secretary problem via robust reductions.

Abstract

We study matroid prophet inequalities when distributions are unknown and accessible only through samples. While single-sample prophet inequalities for special matroids are known, no constant-factor competitive algorithm with even a sublinear number of samples was known for general matroids. Adding more to the stake, the single-sample version of the question for general matroids has close (two-way) connections with the long-standing matroid secretary conjecture. In this work, we give a -competitive matroid prophet inequality with only samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with samples, and (ii) a -selectable matroid OCRS with samples which carefully addresses an adaptivity challenge.
Paper Structure (33 sections, 14 theorems, 26 equations, 1 figure, 6 algorithms)

This paper contains 33 sections, 14 theorems, 26 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Further Related Works
  3. Preliminaries
  4. Prophet Inequalities.
  5. Online Contention Resolution Schemes (OCRSs)
  6. Matroids
  7. Matroid Prophet Inequalities from Samples
  8. Overview of Our Techniques
  9. Our Approach. (Quantile-based.)
  10. Main Difficulty.
  11. Reduction from Matroid Prophet Inequality to OCRS with Samples
  12. Learning Thresholds.
  13. Activation Rule.
  14. Proof of Lemma \ref{['lemma:prophet_competitive_ratio']}
  15. Matroid OCRS from Samples
  16. ...and 18 more sections

Key Result

Theorem 3.1

With $O_\varepsilon(\log^4 n)$ samples, there is a $(\frac{1}{4}-\varepsilon)$-competitive prophet inequality for general matroids of size $n$ and any $\varepsilon>0$.

Figures (1)

  • Figure 1: The hard instance for graphic matroids in HPT22

Theorems & Definitions (16)

  • Theorem 3.1
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 3.6
  • Remark 3.7: Against almighty adversary
  • Lemma 3.8: Lemma 2.4, BFG19
  • Lemma 3.9: Lemma 1, KW12
  • Lemma 3.10
  • Theorem 4.1
  • ...and 6 more