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

TL;DR

This paper improves understanding of three classic algorithms under uncertainty by establishing tight bounds and introducing the Big-Decisions-First principle, which advocates resolving high-stakes decisions early.

Contribution

It resolves a quadratic gap in approximation guarantees for these problems and introduces a unifying principle for decision-making under uncertainty.

Findings

Achieves an $O(\log n)$-approximation for one problem with non-binary values.

Establishes $ ilde{ ext{Omega}}(\log^2 n)$-hardness for two problems.

Unifies the analysis through the Big-Decisions-First principle.

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 $\tilde{\Theta}(\log n)$-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 $\tilde{\Theta}(\log n)$ vs. $\tilde{\Theta}(\log^2 n)$. We resolve this quadratic gap for all three problems, showing $\tilde{\Omega}(\log^2 n)$-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.