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The Competition Complexity of Prophet Inequalities with Correlations

Tomer Ezra, Tamar Garbuz

TL;DR

The results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present.

Abstract

We initiate the study of the prophet inequality problem through the resource augmentation framework in scenarios when the values of the rewards are correlated. Our goal is to determine the number of additional rewards an online algorithm requires to approximate the maximum value of the original instance. While the independent reward case is well understood, we extend this research to account for correlations among rewards. Our results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present. We develop asymptotically optimal algorithms for the following three scenarios: (1) where rewards arrive in blocks corresponding to the different copies of the original instance; (2) where rewards across all copies are arbitrarily shuffled; and (3) where rewards arrive in blocks corresponding to the different copies of the original instance, and values within each block are pairwise independent rather than fully correlated.

The Competition Complexity of Prophet Inequalities with Correlations

TL;DR

The results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present.

Abstract

We initiate the study of the prophet inequality problem through the resource augmentation framework in scenarios when the values of the rewards are correlated. Our goal is to determine the number of additional rewards an online algorithm requires to approximate the maximum value of the original instance. While the independent reward case is well understood, we extend this research to account for correlations among rewards. Our results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present. We develop asymptotically optimal algorithms for the following three scenarios: (1) where rewards arrive in blocks corresponding to the different copies of the original instance; (2) where rewards across all copies are arbitrarily shuffled; and (3) where rewards arrive in blocks corresponding to the different copies of the original instance, and values within each block are pairwise independent rather than fully correlated.
Paper Structure (28 sections, 15 theorems, 49 equations, 2 figures, 1 table, 5 algorithms)

This paper contains 28 sections, 15 theorems, 49 equations, 2 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Structural Contribution.
  4. Quantitative Contribution.
  5. Pairwise Independence.
  6. Our Techniques
  7. Block Arrival Model.
  8. Adversarial Arrival Model.
  9. Pairwise Independence.
  10. Further Related Work
  11. Combinatorial Settings.
  12. Different Arrival Models.
  13. Alternative Benchmarks.
  14. Model
  15. Correlated Prophet Inequality.
  16. ...and 13 more sections

Key Result

Theorem 3.1

Algorithm alg:cap achieves a competition complexity of $(1+o(1)) \cdot (n +\log\log\frac{1}{\varepsilon})$.

Figures (2)

  • Figure 1: In this figure, the black line represents the CDF of the maximum value. The green area above it is the expected value of the prophet.
  • Figure 2: The accumulated value of our algorithm throughout the phases. The red dotted line is the $(1-1/n)$-quantile. The green area is the expected value of the algorithm up to the end of the round of each phase. The pink area (over the total area) is the probability that the algorithm selects some value up to the end of the round of the phase.

Theorems & Definitions (27)

  • Definition 2.1: Competition Complexity
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 17 more