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Gomory-Chvátal cuts are prominent in integer programming. The Gomory-Chvátal closure of a polyhedron is the intersection of all half spaces defined by its Gomory-Chvátal cuts. In this paper, we show that it is NP-complete to decide whether the Gomory-Chvátal closure of a rational polyhedron is empty, even when this polyhedron contains no integer point. This implies that the problem of deciding ...

A famous theorem due to E. Steinitz states, in one of its formulations, that every planar (or, equivalently, every spherical) graph can be realized as the graph of edges and vertices of a convex polyhedron in Euclidean 3-space (see, for example, Grünbaum [1, Section 13.1] or Ziegler [3, Chapter 4]). This representation is possible in many different ways, but in all of them the circuits that bou...

Split cuts form a well-known class of valid inequalities for mixed-integer programming problems. Cook, Kannan and Schrijver (1990) showed that the split closure of a rational polyhedron P is again a polyhedron. In this paper, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We then use this result to prove that cross cuts yield closu...

We considered Turing patterns on a spherical surface from the viewpoint of polyhedron geometry. We restrict our consideration to a set of parameters that produces a pattern of spots. We obtained numerical solutions for the Turing system on a spherical surface and approximated the solutions to convex polyhedrons. The polyhedron structure was dependent on both the radius of the sphere R and the i...

Hybrid systems combine discrete and continuous behavior. We study properties of trajectories of a rectangular hybrid system in which the discrete state goes through a loop. This system is viable if there exists an innnite trajectory starting from some state. We show that the system is viable if and only if it has a limit cycle or xed point. The set of xed points is a polyhedron. The viability k...

A well-known result in the study of convex polyhedra, due to Minkowski, is that a convex polyhedron is uniquely determined (up to translation) by the directions and areas of its faces. The theorem guarantees existence of the polyhedron associated to given face normals and areas, but does not provide a constructive way to find it explicitly. This article provides an algorithm to reconstruct 3D c...

In this paper we tackle the problem of tetrahedralization by breaking non-convex polyhedra into convex subpolyhedra, tetrahedralizing these convex subpolyhedra and merging them together. We generate a Binary Space Partition (BSP) tree from the triangular faces of a polyhedron and use this to identify the convex subpolyhedra in the polyhedron. Each convex subpolyhedron is tetrahedralized individ...

Two problems related to packing identical rectangles within a polyhedron are tackled in the present work. Rectangles are allowed to differ only by horizontal or vertical translations and possibly ninety-degree rotations. The first considered problem consists in packing as many identical rectangles as possible within a given polyhedron, while the second problem consists in finding the smallest p...