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Secretary, Prophet, and Stochastic Probing via Big-Decisions-First

Aviad Rubinstein, Sahil Singla

Abstract

We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight $\tildeΘ(\log n)$-factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional $\log n$ factor when discretizing to binary values, leaving a quadratic gap of $\tildeΘ(\log n)$ vs. $\tildeΘ(\log^2 n)$. We resolve this quadratic gap for all three problems, showing $\tildeΩ(\log^2 n)$-hardness for two of them and an $O(\log n)$-approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early.

Secretary, Prophet, and Stochastic Probing via Big-Decisions-First

Abstract

We revisit three fundamental problems in algorithms under uncertainty: the Secretary Problem, Prophet Inequality, and Stochastic Probing, each subject to general downward-closed constraints. When elements have binary values, all three problems admit a tight -factor approximation guarantee. For general (non-binary) values, however, the best known algorithms lose an additional factor when discretizing to binary values, leaving a quadratic gap of vs. . We resolve this quadratic gap for all three problems, showing -hardness for two of them and an -approximation algorithm for the third. While the technical details differ across settings, and between algorithmic and hardness proofs, all our results stem from a single core observation, which we call the Big-Decisions-First Principle: Under uncertainty, it is better to resolve high-stakes (large-value) decisions early.

Paper Structure

This paper contains 48 sections, 12 theorems, 16 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Worst-case order --- Prophet Inequality.
  4. Best-case order --- Stochastic Probing.
  5. Secretary Problem --- Average-case order.
  6. Related Work
  7. Technical Overview
  8. Secretary Algorithm
  9. Hardness for Prophet Inequality
  10. Large Adaptivity Gap for Stochastic Probing
  11. Single-Log Algorithm for the Secretary Problem
  12. Preliminaries and Preprocessing
  13. Converting to XOS Representation.
  14. Standard Preprocessing.
  15. Setting up the Algorithm
  16. ...and 33 more sections

Key Result

Theorem 1.1

For Prophet Inequality with arbitrary downward-closed constraints and an additive objective, there exist instances where no online algorithm can achieve in expectation better than an $\tilde{\Omega}(\log^2 n)$ approximation. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Prophet inequality construction
  • Figure 2: Caterpillar construction for Stochastic Probing
  • Figure 3: Example of double error-correcting code construction

Theorems & Definitions (30)

  • Theorem 1.1: Informal \ref{['thm:PIHardness']}
  • Theorem 1.2: Informal \ref{['thm:adapGapLB']}
  • Theorem 1.3: Informal \ref{['thm:mainSecretFormal']}
  • Definition 3.1: Secretary Problem
  • Lemma 3.3
  • Claim 3.4
  • proof
  • Theorem 3.5
  • proof
  • proof : Proof of \ref{['eq:singleScale']}
  • ...and 20 more