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

2026-03-31

AI Summary

This work addresses the long-standing $O(\log^2 n)$ approximation gap in the secretary problem, prophet inequalities, and stochastic probing under general downward-closed constraints in non-binary settings. The paper introduces the core principle of “prioritizing large decisions,” which favors high-value elements to better manage uncertainty. By integrating randomized algorithms, discretization techniques, and refined logarithmic factor analysis, the authors develop novel online strategies and adversarial constructions. Their main contributions include the first $O(\log n)$-approximation algorithm for stochastic probing in the non-binary setting, effectively closing the approximation gap, and establishing a matching hardness lower bound of $\widetilde{\Omega}(\log^2 n)$ for both the secretary problem and prophet inequalities, thereby fully resolving this fundamental theoretical gap.

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.

Problem

• Secretary Problem
• Prophet Inequality
• Stochastic Probing
• downward-closed constraints
• approximation gap

Innovation

• Big-Decisions-First
• Secretary Problem
• Prophet Inequality
• Stochastic Probing
• Approximation Algorithms