All 0-1 polytopes are traveling salesman polytopes

We study the facial structure of two important permutation polytopes in R(n2), the Birkhoff or assignment polytope B-n, defined as the convex hull of all n x n permutation matrices, and the asymmetric traveling salesman polytope T-n, defined as the convex hull of those nxn permutation matrices corresponding to n-cycles. Using an isomorphism between the face lattice of B-n and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices of B-n is contained in a cubical face, showing faces of B-n to be fairly special 0-1 polytopes. On the other hand, we show that every 0-1 d-polytope is affinely equivalent to a face of T-n, for d similar to logn, by showing that every 0-1 d-polytope is affinely equivalent to the asymmetric traveling salesman polytope of some directed graph with n nodes. The latter class of polytopes is shown to have maximum diameter [n/3].