All 0-1 Polytopes are Traveling Salesman Polytopes

We study the facial structure of two important permutation polytopes in R n 2 , the Birkho or assignment polytope B n , de ned as the convex hull of all n n permutation matrices, and the asymmetric traveling salesman polytope T n , de ned as the convex hull of those n n 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 T n has every 0-1 d-polytope as a face, for d = logn, by showing that every 0-1 d-polytope is 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 .