Roughly speaking, the rank of a Delaunay polytope (first introduced in [2]) is its number of degrees of freedom. In [3], a method for computing the rank of a Delaunay polytope P using the hypermetrics related to P is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics is given. This method is applied to classical Delaunay polytopes. Then, we give ...

Define the transportation polytope Tn,m to be a polytope of non-negative n×m matrices with row sums equal to m and column sums equal to n. We present a new recurrence relation for the numbers fk of the k-dimensional faces for the transportation polytope Tn,n+1. This gives an efficient algorithm for computing the numbers fk , which solves the problem known to be computationally hard in a general...

For any integral convex polytope in R there is an explicit construction of an error-correcting code of length (q 1) over the nite eld Fq , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimumdistance is estimated using the cohomology and intersection t...

The motivation for this paper is the integer linear programming approach to learning the structure of a decomposable graphical model. We have chosen to represent decomposable models by means of special zero-one vectors, named characteristic imsets. Our approach leads to the study of a special polytope, defined as the convex hull of all characteristic imsets for chordal graphs, named the chordal...

Deene transportation polytope Tn;m to be a polytope of nonnegative n m matrices with row sums equal to m and column sums equal to n. We present an eecient algorithm for computing the numbers f k of the k-dimensional faces for the transportation polytope T n;n+1. The construction relies on the new recurrence relation for the numbers f i , which is of independent interest.

An abstract polytope is flat if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for n > 8, a chiral n-polytope has at least 48(n− 2)(n− 2)! flags.

Abstract. We study the Chvátal-rank of two linear relaxations of the stable set polytope, the edge constraint and the clique constraint stable set polytope. For some classes of graphs whose stable set polytope is given by 0/1-valued constraints only, we present either the exact value of the Chvátal-rank or upper bounds (of the order of their largest cliques and stable sets) which improve the bo...

We study Lovász and Schrijver’s hieararchy of relaxations based on positive semidefiniteness constraints derived from the fractional stable set polytope. We show that there are graphsG for which a single application of the underlying operator, N+, to the fractional stable set polytope gives a nonpolyhedral convex relaxation of the stable set polytope. We also show that none of the current best ...