Faces of Birkhoff Polytopes

The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation matrices, i.e., matrices where precisely one entry in each row and column is one, and zeros at all other places. This is a widely studied polytope with various applications throughout mathematics. In this paper we study combinatorial types L of faces of a Birkhoff polytope. The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face with combinatorial type L. By a result of Billera and Sarangarajan, a combinatorial type L of a d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is at most d. We will characterize those types whose Birkhoff dimension is at least 2d-3, and we prove that any type whose Birkhoff dimension is at least d is either a product or a wedge over some lower dimensional face. Further, we computationally classify all d-dimensional combinatorial types for d between 2 and 8.Comment: 29 page

Magic graphs and the faces of the Birkhoff polytope

Magic labelings of graphs are studied in great detail by Stanley and Stewart. In this article, we construct and enumerate magic labelings of graphs using Hilbert bases of polyhedral cones and Ehrhart quasi-polynomials of polytopes. We define polytopes of magic labelings of graphs and digraphs. We give a description of the faces of the Birkhoff polytope as polytopes of magic labelings of digraphs.Comment: 9 page

Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

A Basis for Slicing Birkhoff Polytopes

We present a change of basis that may allow more efficient calculation of the volumes of Birkhoff polytopes using a slicing method. We construct the basis from a special set of square matrices. We explain how to construct this basis easily for any Birkhoff polytope, and give examples of its use. We also discuss possible directions for future work