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**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, several 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 that approximates the optimal online algorithm strictly better than the best-possible prophet inequality. In contrast, we show that it is PSPACE-hard to approximate this problem within some constant smaller than 1.

Joint work with Christos Papadimitriou, Tristan Pollner, and David Wajc

**Bio:** Amin Saberi is a Professor of Management Science and Engineering at Stanford University. He received his B.Sc. from Sharif University of Technology and his Ph.D. from Georgia Institute of Technology in Computer Science. His research interests include algorithms and applications in market design and analysis. He is a recipient of the Terman Fellowship, Alfred Sloan Fellowship, and several best paper awards.

Amin was the founding CEO and chairman of NovoEd, a social learning environment designed in his research lab and used by universities such as Stanford, UC Berkeley, University of Michigan, and several non-profit and for-profit institutions. He is currently advising the marketplace and data and applied science teams at Uber on pricing and matching algorithms.