On matroid parity and matching polytopes

Abstract

The matroid parity (MP) problem is a powerful (and -hard) extension of the matching problem. Whereas matching polytopes are well understood, little is known about MP polytopes. We prove that, when the matroid is laminar, the MP polytope is affinely congruent to a perfect -matching polytope. From this we deduce that, even when the matroid is not laminar, every Chvátal–Gomory cut for the MP polytope can be derived as a -cut from a laminar family of rank constraints. We also prove a negative result concerned with the integrality gap of two linear relaxations of the MP problem.