A compact linear program for testing optimality of perfect matching

It is a longstanding
open problem whether there exists a polynomial size description
of the perfect matching polytope.
We give a partial answer to this question by proving
the following result. The polyhedron defined by the constraints of
the perfect matching polytope which are active at a given perfect
matching can be obtained as the projection of a compact polyhedron.
Thus there exists a compact linear program which is
unbounded if and only if the perfect matching is not optimal with
respect to a given edge weight. This result provides a simple
reduction of the maximum weight perfect matching problem to compact
linear programming.