Single-Sample Prophet Inequalities: A Combinatorial to Single-Item Reduction

2026-10-02

AI Summary

This study addresses the theoretical bottleneck of single-sample prophet inequalities in online combinatorial allocation. It proposes a modular reduction framework based on supporting prices that decouples combinatorial constraints from supply, enabling a general reduction from combinatorial to single-item prophet inequalities. This approach extends to the stronger Googol game model and integrates techniques including free disposal, XOS valuation analysis, and online algorithm design. By overcoming existing theoretical limitations in combinatorial allocation, this work achieves a competitive ratio of approximately 1/10.4 for the single-sample setting and β/4 for the k-sample setting, while attaining optimal solutions for fractional knapsack problems. Ultimately, these results establish a new paradigm for online mechanism design.

Abstract

We study single-sample prophet inequalities for online combinatorial allocation. Our main contribution is a general reduction from combinatorial to single-item prophet inequalities for valuation classes admitting suitable supporting prices. The reduction uses a free-disposal value to separate buyer-side combinatorial constraints from item-side supply constraints, yielding a modular framework that applies in the stronger Game of Googol model. This framework yields a $\frac{1}{6\sqrt{3}}\approx\frac{1}{10.4}$-competitive single-sample prophet inequality and a $(β_{k-1}/4)$-competitive $k$-sample prophet inequality for XOS valuations, where $β_k$ is the competitive ratio of a $k$-sample single-item prophet inequality, improving upon the work of [DKL+24]. Both results extend directly to divisible resources with capped-XOS valuations. Along the way, we obtain new results for online free disposal and an optimal single-sample prophet inequality for fractional knapsack in the Game of Googol model.

Problem

• single-sample prophet inequalities
• online combinatorial allocation
• Game of Googol model
• XOS valuations
• competitive ratio

Innovation

• Single-Sample Prophet Inequalities
• Combinatorial Allocation
• Reduction Framework
• XOS Valuations
• Game of Googol