  It is well-known that the intersection of the matching polytope with a
cardinality constraint is integral [8]. We prove a similar result for the
polytope corresponding to the transportation problem with market choice (TPMC)
(introduced in [4]) when the demands are in the set $\{1,2\}$. This result
generalizes the result regarding the matching polytope and also implies that
some special classes of minimum weight perfect matching problem with a
cardinality constraint on a subset of edges can be solved in polynomial time.
