We show that given two vertices of a polytope one cannot in general nd a hyperplane containing the vertices, that has two or more facets of the polytope in one closed half-space. Our result refutes a long-standing conjecture. We prove the result by constructing a 4-dimensional polytope that provides the counterexample. Also, we show that such a cutting hyperplane can be found for each pair of v...

Reference implementations of signal processing applications are often written in a sequential language that does not reveal the available parallelism in the application. However, if an application satisfies some constraints then a parallel specification can be derived automatically. In particular, if the application can be represented in the polyhedral model, then a polyhedral process network c...

The secondary polytope of a point configuration A is a polytope whose face poset is isomorphic to the poset of all regular subdivisions of A. While the vertices of the secondary polytope – corresponding to the triangulations of A – are very well studied, there is not much known about the facets of the secondary polytope. The splits of a polytope, subdivisions with exactly two maximal faces, are...

We investigate the family of facet defining inequalities for the asymmetric traveling salesman (ATS) polytope obtainable by lifting the cycle inequalities. We establish several properties of this family that earmark it as the most important among the asymmetric inequalities for the ATS polytope known to date: (i) The family is shown to contain members of unbounded Chvatal rank, whereas most kno...

We study finite groups that occur as combinatorial automorphism or geometric symmetry of convex polytopes. When $\Gamma$ is a subgroup the group $d$-polytope, $d\geq 3$, then there exists $d$-polytope related to original polytope with exactly $\Gamma$. both and These symmetry-breaking results are applied show for every abelian even order involution $\sigma$ $\Gamma$, centrally symmetric such co...

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. study its interplay with general presentation for any finite-dimensional basic algebra. give an interpretation evaluating at weight vector. As consequence, we Newton polytope ${\sf N}(M)$ dual fan and 1-skeleton N}(M)$. propose algorithm to determine generic polytopes, show it works path algebras. application, repres...

Proofs of these relations may be found in [1, 2, 3]. If the polytope ∆ is not simple, then the relations above are not true. A polytope ∆ is said to be integral provided all its vertices belong to the integer lattice. With each integral convex polytope ∆ one associates the toric variety X = X(∆) (see [4, 5, 6, 7]). This is a projective complex algebraic variety, singular in general. It turns ou...

Given a polyhedron P which is of interest, a major goal of polyhedral combinatorics is to find classes of essential, i.e. facet inducing inequalities which describe P . In general this is a difficult task. We consider the case in which we have knowledge of facets for a face F of P , and present some general theory and methods for exploiting the close relationship between such polyhedra in order...

The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm’s rate of convergence depends on the radius of the largest ball around the orig...

This thesis deals with the multi-terminal vertex separator problem. Given a graph G = (V ∪T,E) with V ∪T the set of vertices, where T is a set of terminals, and a weight function w : V → Z, associated with nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V ∪T into k+1 subsets {S, V1, . . . , Vk} such that there is no edge between two different subsets Vi a...