  The cutting plane approach to optimal matchings has been discussed by several
authors over the past decades (e.g., Padberg and Rao '82, Grotschel and Holland
'85, Lovasz and Plummer '86, Trick '87, Fischetti and Lodi '07) and its
convergence has been an open question. We give a cutting plane algorithm that
converges in polynomial-time using only Edmonds' blossom inequalities; it
maintains half-integral intermediate LP solutions supported by a disjoint union
of odd cycles and edges. Our main insight is a method to retain only a subset
of the previously added cutting planes based on their dual values. This allows
us to quickly find violated blossom inequalities and argue convergence by
tracking the number of odd cycles in the support of intermediate solutions.
