Prophet Inequalities via Linear Programming

2026-02-07

AI Summary

This work investigates how computable online strategies can approximate the offline optimal expected reward in prophet inequality problems. The authors introduce a unified framework by systematically applying linear programming to the analysis of prophet inequalities, integrating tools from polyhedral theory, matroid and submodular optimization, and Minkowski sum analysis. Their approach, centered on a simplified linear program, not only recovers several classical results but also establishes a 1/2-approximation guarantee for the online matroid setting. Furthermore, the study reveals that the Minkowski sum operation preserves the combinatorial structure underlying prophet inequality constants, thereby offering new theoretical tools and analytical perspectives for related problems in online decision-making.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

Prophet inequalities bound the expected reward that can be obtained in a stopping problem by the optimal reward of its corresponding off-line version. We propose a systematic technique for deriving prophet inequalities for stopping problems associated with selecting a point in a polyhedron. It utilizes a reduced-form linear programming representation of the stopping problem. We illustrate the technique to derive a number of known results as well as some new ones. For instance, we prove a $\frac{1}{2}$-prophet inequality when the underlying polyhedron is an on-line polymatroid; one whose underlying submodular function depends upon the realized rewards. We also demonstrate a composition by the Minkowski sum property. If an $r-$ prophet inequality holds for polyhedra $P^1$ and $P^2$, it also holds for their Minkowski sum.

Problem

• prophet inequalities
• stopping problems
• polyhedra
• online polymatroid
• Minkowski sum

Innovation

• prophet inequalities
• linear programming
• stopping problems
• polymatroid
• Minkowski sum