  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 $\alpha < 1$.
