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?

Motivated by applications in two-sided matching markets, such as housing markets, 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 α < 1.

Based on joint work with Christos Papadimitriou, Tristan Pollner and Amin Saberi.