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Pairwise-Independent Contention Resolution

Anupam Gupta, Jinqiao Hu, Gregory Kehne, Roie Levin

TL;DR

These results complement recent work of Dughmi, Kalayci, and Patel (STOC '24) showing that no $\omega(1/k)$-selectable OCRS exists in the PI setting for general matroids of rank $k$.

Abstract

We study online contention resolution schemes (OCRSs) and prophet inequalities for non-product distributions. Specifically, when the active set is sampled according to a pairwise-independent (PI) distribution, we show a $(1-o_k(1))$-selectable OCRS for uniform matroids of rank $k$, and $Ω(1)$-selectable OCRSs for laminar, graphic, cographic, transversal, and regular matroids. These imply prophet inequalities with the same ratios when the set of values is drawn according to a PI distribution. Our results complement recent work of Dughmi, Kalayci, and Patel (STOC '24) showing that no $ω(1/k)$-selectable OCRS exists in the PI setting for general matroids of rank $k$.

Pairwise-Independent Contention Resolution

TL;DR

These results complement recent work of Dughmi, Kalayci, and Patel (STOC '24) showing that no -selectable OCRS exists in the PI setting for general matroids of rank .

Abstract

We study online contention resolution schemes (OCRSs) and prophet inequalities for non-product distributions. Specifically, when the active set is sampled according to a pairwise-independent (PI) distribution, we show a -selectable OCRS for uniform matroids of rank , and -selectable OCRSs for laminar, graphic, cographic, transversal, and regular matroids. These imply prophet inequalities with the same ratios when the set of values is drawn according to a PI distribution. Our results complement recent work of Dughmi, Kalayci, and Patel (STOC '24) showing that no -selectable OCRS exists in the PI setting for general matroids of rank .
Paper Structure (29 sections, 29 theorems, 90 equations)

This paper contains 29 sections, 29 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Preliminaries
  5. The Almighty Adversary
  6. The Offline Adversary
  7. Prophet Inequalities and OCRSs
  8. Uniform Matroids
  9. A $(1-O(k^{-1/5}))$-Selectable PI-OCRS for Uniform Matroids
  10. Laminar Matroids
  11. Graphic Matroids
  12. Transversal Matroids
  13. Cographic Matroids
  14. Regular Matroids
  15. The Dinitz-Kortsarz modification.
  16. ...and 14 more sections

Key Result

Theorem 1.1

There is an algorithm in the prophet model for $k$-uniform matroids that achives expected value at least $(1-O(k^{-1/5}))$ of the expected optimal value.

Theorems & Definitions (75)

  • Theorem 1.1: Uniform Matroid PI Prophets
  • Theorem 1.2: Other Matroids PI Prophets
  • Theorem 1.3: Upper Bound for Multiple Thresholds
  • Theorem 1.4: Single Sample Prophet Inequality
  • Theorem 1.5: $t$-wise Independent Prophet Inequality
  • Definition 2.1: PI-consistency
  • Claim 2.2
  • Definition 2.3: $(b,c)$-balanced PI-CRS
  • Claim 2.4
  • proof
  • ...and 65 more