  We settle the complexities of the maximum-cardinality bipartite matching
problem (BMM) up to poly-logarithmic factors in five models of computation: the
two-party communication, AND query, OR query, XOR query, and quantum edge query
models. Our results answer open problems that have been raised repeatedly since
at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck,
Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan
SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21]
and Zhang [ICALP'04]. We also settle the communication complexity of the
generalizations of BMM, such as maximum-cost bipartite $b$-matching and
transshipment; and the query complexity of unique bipartite perfect matching
(answering an open question by Beniamini [2022]). Our algorithms and lower
bounds follow from simple applications of known techniques such as cutting
planes methods and set disjointness.
