We study the facial structure of two important permutation polytopes in IR n2 , the Birkhoff or assignment polytope Bn, defined as the convex hull of all n x n permutation matrices, and the asymmetric traveling salesman polytope Tn, defined as the convex hull of those nxn permutation matrices corresponding to n-cycles. Using an isomorphism between the face lattice of Bn and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices of Bn is contained in a cubical face, showing faces of Bn 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 Tn, for d~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 [-~ J.
Abstract. We provide lower and upper bounds for the maximal number of facets of a d-dimensional 0/1-polytope, and for the maximal number of vertices that can appear in a two-dimensional projection ("shadow") of such a polytope.
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