Single-Sample Prophet Inequalities via Greedy-Ordered Selection
Constantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, Rebecca Reiffenhäuser
TL;DR
The paper introduces a versatile greedy-ordered framework for direct single-sample prophet inequalities (SSPI), bypassing the prior heavy reliance on reductions to order-oblivious secretary policies. By extracting thresholds from a sample-driven greedy pass and then online selecting elements whose rewards exceed these thresholds while preserving feasibility, the authors derive strong SSPIs across several central problems, including general and bipartite matching, budget-additive combinatorial auctions, and matroids, achieving new constants such as $16$, $8$, and $24$-competitive guarantees. They also provide truthful mechanisms in the bipartite setting and extend the framework to transversal matroids, while analyzing fundamental limits via a partial converse showing that certain SSPIs imply corresponding OOS policies. A meta-theorem for α-partition matroids yields a generic, efficient reduction yielding a $2α$-competitive SSPI, highlighting the reach of direct SSPIs. Finally, the work connects SSPIs to P-SSPIs and OOS, offering a deeper understanding of when constant-factor SSPIs are possible and illustrating the landscape between SSPIs, OOS reductions, and full-information prophet inequalities, with broad implications for online mechanism design and scalable approximation algorithms.
Abstract
We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy, reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.