We investigate geometric and topological properties of $d$-majorization -- a generalization classical majorization to positive weight vectors $d \in \mathbb{R}^n$. In particular, we derive new, simplified characterization which allows us work out halfspace description the corresponding polytopes. That is, write set all are $d$-majorized by some given vector $y \mathbb{R}^n$ as an intersection f...

We consider three well-studied polyhedral relaxations for the maximum cut problem: the metric polytope of the complete graph, the metric polytope of a general graph, and the relaxation of the bipartite subgraph polytope. The metric polytope of the complete graph can be described with a polynomial number of inequalities, while the latter two may require exponentially many constraints. We give an...

The max-cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years. However, little research has been conducted for the cut polytope on arbitrary graphs. In this study we describe new separation and lifting procedures for the cut polytope on such graphs. These procedures exploit algorithmic and structural results known for the cut polytop...

A sweep-plane algorithm of Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller...

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 Rothvoÿ [2011] it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}) such that any higherdimensional polytope projecting to it must have 2 facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high PSD extension complexity was left open. We answer this question in the a rmative by showing that the...

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...

In this paper we introduce two polytopes that respect a digraph in the sense that for every vector in the polytope every component corresponds to a node and is at least equal to the component corresponding to each successor of this node. The sharing polytope is the set of all elements from the unit simplex that respect the digraph. The fuzzy polytope is the set of all elements of the unit cube ...

The vertices of the secondary polytope of a point connguration correspond to its regular triangulations. The Cayley trick links triangulations of one point connguration, called the Cayley polytope, to the ne mixed subdivisions of a tuple of point conngurations. In this paper we investigate the secondary polytope of this Cayley polytope. Its vertices correspond to all regular mixed subdivisions ...