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Polyhedral representation conversion up to symmetries
, 2009
"... We give a short survey on computational techniques which can be used to solve the representation conversion problem for polyhedra up to symmetries. In particular we discuss decomposition methods, which reduce the problem to a number of lower dimensional subproblems. These methods have been successfu ..."
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We give a short survey on computational techniques which can be used to solve the representation conversion problem for polyhedra up to symmetries. In particular we discuss decomposition methods, which reduce the problem to a number of lower dimensional subproblems. These methods have been successfully used by different authors in special contexts. Moreover, we sketch an incremental method, which is a generalization of Fourier–Motzkin elimination, and we give some ideas how symmetry can be exploited using pivots.
Which nonnegative matrices are slack matrices?
, 2013
"... In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown. ..."
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In this paper we characterize the slack matrices of cones and polytopes among all nonnegative matrices. This leads to an algorithm for deciding whether a given matrix is a slack matrix. The underlying decision problem is equivalent to the polyhedral verification problem whose complexity is unknown.
A New Approach to OutputSensitive Voronoi Diagrams
"... We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest ..."
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We describe a new algorithm for computing the Voronoi diagram of a set of n points in constantdimensional Euclidean space. The running time of our algorithm is O(f log n log ∆) where f is the output complexity of the Voronoi diagram and ∆ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and nearlinear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures. 1
SUM OF SQUARES CERTIFICATES FOR CONTAINMENT OF HPOLYTOPES IN VPOLYTOPES
"... Abstract. Given an Hpolytope P and a Vpolytope Q, the decision problem whether P is contained in Q is coNPcomplete. This hardness remains if P is restricted to be a standard cube and Q is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orli ..."
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Abstract. Given an Hpolytope P and a Vpolytope Q, the decision problem whether P is contained in Q is coNPcomplete. This hardness remains if P is restricted to be a standard cube and Q is restricted to be the affine image of a cross polytope. While this hardness classification by Freund and Orlin dates back to 1985, for general dimension there seems to be only limited progress on that problem so far. Based on a formulation of the problem in terms of a bilinear feasibility problem, we study sum of squares certificates to decide the containment problem. These certificates can be computed by a semidefinite hierarchy. As a main result, we show that under mild and explicitly known preconditions the semidefinite hierarchy converges in finitely many steps. In particular, if P is contained in a large Vpolytope Q (in a welldefined sense), then containment is certified by the first step of the hierarchy. 1.