On Matroid Parity and Matching Polytopes∗

The matroid parity (MP) problem is a natural extension of the matching problem to the matroid setting. It can be formulated as a 0− 1 linear program using the so-called rank and line constraints. We call the associated family of polytopes MP polytopes. We then prove the following: (i) when the matroid is a gammoid, each MP polytope is a projection of a perfect matching polytope into a suitable subspace; (ii) when the matroid is laminar, each MP polytope is affinely congruent to a perfect matching polytope; (iii) even if the matroid is laminar, MP polytopes can have facets that are defined by inequalities with nonternary left-hand side coefficients; (iv) for any matroid, the elementary closure of the continuous relaxation of the rank-and-line formulation is equal to its {0, 12}-closure.