9,021 research outputs found
Asymmetric Convex Intersection Testing
We consider asymmetric convex intersection testing (ACIT).
Let P subset R^d be a set of n points and H a set of n halfspaces in d dimensions. We denote by {ch(P)} the polytope obtained by taking the convex hull of P, and by {fh(H)} the polytope obtained by taking the intersection of the halfspaces in H. Our goal is to decide whether the intersection of H and the convex hull of P are disjoint. Even though ACIT is a natural variant of classic LP-type problems that have been studied at length in the literature, and despite its applications in the analysis of high-dimensional data sets, it appears that the problem has not been studied before.
We discuss how known approaches can be used to attack the ACIT problem, and we provide a very simple strategy that leads to a deterministic algorithm, linear on n and m, whose running time depends reasonably on the dimension d
A computational comparison of two simplicial decomposition approaches for the separable traffic assignment problems : RSDTA and RSDVI
Draft pel 4th Meeting del Euro Working Group on Transportation (Newcastle 9-11 setembre de 1.996)The class of simplicial decomposition methods has shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. The paper presents a particular implementation of such an algorithm, called RSDVI, and a restricted simplicial decomposition algorithm, developed adhoc for diagonal, separable, problems named RSDTA. Both computer codes are compared for large scale separable traffic assignment problems. Some meaningful figures are shown for general problems with several levels of asymmetry.Preprin
The computational complexity of convex bodies
We discuss how well a given convex body B in a real d-dimensional vector
space V can be approximated by a set X for which the membership question:
``given an x in V, does x belong to X?'' can be answered efficiently (in time
polynomial in d). We discuss approximations of a convex body by an ellipsoid,
by an algebraic hypersurface, by a projection of a polytope with a controlled
number of facets, and by a section of the cone of positive semidefinite
quadratic forms. We illustrate some of the results on the Traveling Salesman
Polytope, an example of a complicated convex body studied in combinatorial
optimization.Comment: 24 page
Graph isomorphism and volumes of convex bodies
We show that a nontrivial graph isomorphism problem of two undirected graphs,
and more generally, the permutation similarity of two given
matrices, is equivalent to equalities of volumes of the induced three convex
bounded polytopes intersected with a given sequence of balls, centered at the
origin with radii , where is an increasing
sequence converging to . These polytopes are characterized by
inequalities in at most variables. The existence of fpras for computing
volumes of convex bodies gives rise to a semi-frpas of order at
most to find if given two undirected graphs are isomorphic.Comment: 9 page
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
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