Prophet Inequalities for Matching with a Single Sample

TL;DR

This paper develops single-sample prophet inequalities for matching problems, providing constant-factor approximations under limited distribution access, and applies these results to design truthful mechanisms for online bipartite matching.

Contribution

It introduces the first single-sample prophet inequalities for matching with general graphs and bipartite graphs, improving approximation factors and enabling truthful mechanism design.

Findings

16-approximate prophet inequality for general graphs with a single sample

8-approximate prophet inequality for bipartite matching with a single sample

Conversion of prophet inequalities into truthful mechanisms for online bipartite matching

Abstract

We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $8$ that also uses just a single sample from each distribution. Finally, we show how to turn our $16$-approximate single-sample prophet inequality into a truthful single-sample mechanism for online bipartite matching with vertex arrivals.