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The cutting plane approach to optimal matchings has been discussed by severalauthors 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 itsconvergence has been an open question. We give a cutting plane algorithm thatconverges in polynomial-time using only Edmonds' blossom inequalities; itmaintains half-integral intermediate LP solutions supported by a disjoint unionof odd cycles and edges. Our main insight is a method to retain only a subsetof the previously added cutting planes based on their dual values. This allowsus to quickly find violated blossom inequalities and argue convergence bytracking the number of odd cycles in the support of intermediate solutions.

The Cutting Plane Method is Polynomial for Perfect Matchings