On the even permutation polytope

On the Even Permutation Polytope

Some Facets of the Polytope of Even Permutation Matrices

We describe a class of facets of the polytope of convex combinations of the collection of even n × n permutation matrices. As a consequence, we prove the conjecture of Brualdi and Liu [4] that the number of facets of the polytope is not bounded by a polynomial in n.

The polytope of even doubly stochastic matrices

The polytope Q, of the convex combinations of the permutation matrices of order n is well known (Birkhoff’s theorem) to be the polytope of doubly stochastic matrices of order n. In particular it is easy to decide whether a matrix of order n belongs to Q,. . check to see that the entries are nonnegative and that all row and columns sums equal 1. Now the permutations z of { 1, 2, . . . . n} are i...

Beyond the Birkhoff Polytope: Convex Relaxations for Vector Permutation Problems

The Birkhoff polytope (the convex hull of the set of permutation matrices), which is represented using Θ(n) variables and constraints, is frequently invoked in formulating relaxations of optimization problems over permutations. Using a recent construction of Goemans [1], we show that when optimizing over the convex hull of the permutation vectors (the permutahedron), we can reduce the number of...

The k-assignment Polytope, Phylogenetic Trees, and Permutation Patterns

In this thesis three combinatorial problems are studied in four papers. In Paper 1 we study the structure of the k-assignment polytope, whose vertices are the m× n (0,1)-matrices with exactly k 1:s and at most one 1 in each row and each column. This is a natural generalisation of the Birkhoff polytope and many of the known properties of the Birkhoff polytope are generalised. A representation of...