Some facets of the polytope of even permutation matrices

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.

On the Even Permutation Polytope

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...

The Circuit Polytope: Facets

Given an undirected graph G = (V;E) and a cost vector c 2 IRE , the weighted girth problem is to nd a circuit in G having minimum total cost. This problem is in general NP-hard since the traveling salesman problem can be reduced to it. A promising approach to hard combinatorial optimization problems is given by the so-called cutting plane methods. These involve linear programming techniques bas...

extensions of some polynomial inequalities to the polar derivative

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