Christophe Weibel

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The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes called perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of faces of the sum in terms of the face lattice of a given(More)
Consider a <i>routing problem</i> instance consisting of a <i>demand graph H</i> = (<i>V, E</i>(<i>H</i>)) and a <i>supply graph G</i> = (<i>V, E</i>(<i>G</i>)). If the pair obeys the cut condition, then the <i>flow-cut gap</i> for this instance is the minimum value <i>C</i> such that there exists a feasible multiflow for <i>H</i> if each edge of <i>G</i>(More)
It is known that in the Minkowski sum of r polytopes in dimension d, with r < d, the number of vertices of the sum can potentially be as high as the product of the number of vertices in each summand [2]. However, the number of vertices for sums of more polytopes was unknown so far. In this paper, we study sums of polytopes in general orientations, and show(More)
We present a tight bound on the exact maximum complexity of Minkowski sums of convex polyhedra in R 3. In particular , we prove that the maximum number of facets of the Minkowski sum of two convex polyhedra with m and n facets respectively is bounded from above by f (m, n) = 4mn−9m−9n+26. Given two positive integers m and n, we describe how to construct two(More)
For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show(More)
The 3D visibility skeleton is a data structure used to encode global visibility information about a set of objects. Previous theoretical results have shown that for k convex polytopes with n edges in total, the worst case size complexity of this data structure is Θ(n 2 k 2) [Brönnimann et al. 07]; whereas for k uniformly distributed unit spheres, the(More)
We present an implementation of a reverse-search algorithm of Fukuda for computing Minkowski sums of polytopes efficiently. The algorithm allows summing any number of polytopes in any dimension, and is complete in the sense that it does not assume general position. Its running time depends linearly on the size of the output. To the best of our knowledge,(More)
Let G=(V,E) be a supply graph and H=(V,F) a demand graph defined on the same set of vertices. An assignment of capacities to the edges of G and demands to the edges of H is said to satisfy the <i>cut condition</i> if for any cut in the graph, the total demand crossing the cut is no more than the total capacity crossing it. The pair (G,H) is called(More)