  Online bipartite matching (OBM) is a fundamental model underpinning many
important applications, including search engine advertisement, website banner
and pop-up ads, and ride-hailing. We study the i.i.d. OBM problem, where one
side of the bipartition is fixed and known in advance, while nodes from the
other side appear sequentially as i.i.d. realizations of an underlying
distribution, and must immediately be matched or discarded. We introduce
dynamic relaxations of the set of achievable matching probabilities, show how
they theoretically dominate lower-dimensional, static relaxations from previous
work, and perform a polyhedral study to theoretically examine the new
relaxations' strength. We also discuss how to derive heuristic policies from
the relaxations' dual prices, in a similar fashion to dynamic resource prices
used in network revenue management. We finally present a computational study to
demonstrate the empirical quality of the new relaxations and policies.
