A proof for Padberg's conjecture on rank of matching polytope

Padberg [2] introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true. 1 Preliminaries Padberg [2] defined the notion of geometric ranks for the facets of the polytopes. All definitions provided in this section could be found in [2]. Let the following be the set of feasible solutions to an integer linear program. Q = {x ∈ Z : Ax ≤ b} We can describe P = conv(Q) (convex hull of Q) by the solution set of ideal, i.e. a minimal linear description as follows. P = {x ∈ R : H1x = h1, H2x ≤ h2}, where (H1, h1) is s× (n+ 1) full rank matrix. Thus the dimension of P is dim(P ) = n− s. (H2, h2) is t × (n + 1) matrix of rationals for some integer t. The rows of the matrix defines a facet of P . Let F be the set of row vectors (h, h0) ∈ R from the matrix (H2, h2). Each of these inequalities induces a facet of P . For a (f, f0) ∈ F , the facet Ff = {x ∈ P : fx = f0} is a polyhedron of dimension dim(P )− 1. In order to define the notion of ranks, Padberg [2] required the notion of facet of facets, i.e., the ridges of P . Let H = {(h, h0) ∈ F : dim(Ff ∩ Fh) = dim(P )− 2} [email protected] [email protected]