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Prophet Secretary and Matching: the Significance of the Largest Item

Ziyun Chen, Zhiyi Huang, Dongchen Li, Zhihao Gavin Tang

Abstract

The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a $0.688$-competitive algorithm, that breaks the $0.675$ barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a $0.641$-competitive algorithm for the prophet secretary matching problem, that breaks the $1-1/e\approx 0.632$ barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).

Prophet Secretary and Matching: the Significance of the Largest Item

Abstract

The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a -competitive algorithm, that breaks the barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a -competitive algorithm for the prophet secretary matching problem, that breaks the barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).

Paper Structure

This paper contains 46 sections, 34 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Result for Prophet Secretary.
  4. Technique: Activation-Based Algorithms.
  5. Technique: Significance of the Largest Item.
  6. Result for Prophet Secretary Matching.
  7. Summary of Techniques for Prophet Secretary Matching.
  8. Related Works
  9. Prophet Secretary
  10. Model
  11. Continuous Arrival Time vs. Discrete Arrival Order.
  12. Our Result
  13. Activation-Based Online Algorithms
  14. General Framework
  15. Activation Rates
  16. ...and 31 more sections

Key Result

Theorem 1

[theorem]thm:prophet-secretary There exists a $0.688$-competitive algorithm for prophet secretary.

Figures (4)

  • Figure 1: Algorithm for Prophet Secretary
  • Figure 2: Competitive Ratio Curves For MAM, CAR, Hybrid
  • Figure 3: Illustration of intervals $I_{u_1},I_{u_2}$ and $I_{u_3}$for $v_i=v$.
  • Figure 4: A Plot of $F_1(h) = c_1(h)+c_2(h)+c_3(h)$ and $F_2(h) = c_2(h)+(2-e^{-(\beta_1-\beta_0)}) c_3(h)$.

Theorems & Definitions (58)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 48 more