Online Stochastic Max-Weight Bipartite Matching: Beyond Prophet Inequalities

TL;DR

This paper introduces a polynomial-time algorithm that approximates the optimal online stochastic maximum-weight bipartite matching within a factor of 0.51, surpassing the traditional prophet inequality benchmark, and discusses the problem's computational hardness.

Contribution

It presents the first polynomial-time algorithm achieving a 0.51-approximation for the optimal online matching, exceeding the prophet inequality ratio, and establishes PSPACE-hardness for better approximations.

Findings

Polynomial-time algorithm with 0.51 approximation ratio.

Surpasses the 1/2 prophet inequality benchmark.

Proves PSPACE-hardness for approximations better than a certain constant.

Abstract

The rich literature on online Bayesian selection problems has long focused on so-called prophet inequalities, which compare the gain of an online algorithm to that of a "prophet" who knows the future. An equally-natural, though significantly less well-studied benchmark is the optimum online algorithm, which may be omnipotent (i.e., computationally-unbounded), but not omniscient. What is the computational complexity of the optimum online? How well can a polynomial-time algorithm approximate it? We study the above questions for the online stochastic maximum-weight matching problem under vertex arrivals. For this problem, a number of $1/2$-competitive algorithms are known against optimum offline. This is the best possible ratio for this problem, as it generalizes the original single-item prophet inequality problem. We present a polynomial-time algorithm which approximates the optimal online algorithm within a factor of $0.51$ -- beating the best-possible prophet inequality. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant $\alpha < 1$.