In the paper we investigate the linear ordering polytope P n. We consider the 3-dicycle relaxation polytope B n and give an example of a fractional vertex of B n such that all non-integral values of the vertex are not equal to 1=2. In this paper also we present a constructive operation (5-Extension) for generating new facets of the linear ordering polytope P n. By connecting MM obius ladders wi...

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain. 1991 Mathematics Subject classification: primary 52C07; secondary 11H55 An n-dimensional lattice determines two normal partitions of the ...

A (convex) polytope is the convex hull of finitely many points in Rd. Alternatively, a polytope can be defined as the bounded intersection of finitely many halfspaces, that is, a polytope is the (bounded) solution set of a system of linear equations and inequalities. (This latter fact is the main non-geometric reason why people are interested in polytopes—linear systems are everywhere.) The mai...

A code polytope is defined to be the convex hull in R n of the points in {0, 1}n corresponding to the codewords of a binary linear code. This paper contains a collection of results concerning the structure of such code polytopes. A survey of known results on the dimension and the minimal polyhedral representation of a code polytope is first presented. We show how these results can be extended t...

We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdière matrices and the Maxwell–Cremona lifting method. We show that every truncated 3d polytope with n vertices can be realized on a grid of size O(n). Moreove...

In this paper 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 the faces by certain bipartite graphs is given. This tool is used to...

In this work we present a procedure for automatic parallel code generation in the case of algorithms described through Set of Affine Recurrence Equations (SARE); starting from the original SARE description in an N-dimensional iteration space, the algorithm is converted into a parallel code for an m-dimensional distributed memory parallel machine (m < N). The used projection technique is based o...

In this paper, we show that the Newton polytope of an observation Y from a two-dimensional hidden Markov model (2D HMM) lies in a three-dimensional subspace of its ambient eight-dimensional space, whose vertices correspond to the most likely explanations (“hidden” states) for Y given the model. For each Newton polytope, there exists a set of “essential” vertices, which form a skeleton for the p...

We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has 2Ω(n) vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k ≥ 2r+1 and every cut in the g...