# Algorithms for polytope covering and approximation

Conference paper

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## Abstract

This paper gives an algorithm for , where γγ1/[

*polytope covering*: let*L*and*U*be sets of points in*R*^{d}, comprising*n*points altogether. A cover for*L*from*U*is a set*C⊂U*with*L*a subset of the convex hull of*C*. Suppose*c*is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than*c*(5*d*ln*c*), for*c*large enough. The algorithm requires*O*(*c*^{2}*n*^{1+δ}) expected time. More exactly, the time bound is$$O(cn^{1 + \delta } + c(nc)^{1/(1 + \gamma /(1 + \delta ))} )$$

*d*/2]. The previous best bounds were*cO*(log*n*) cover size in*O(n*^{d}) time.[MS92b] A variant algorithm is applied to the problem of approximating the boundary of a polytope with the boundary of a simpler polytope. For an appropriate measure, an approximation with error*ε*requires c=O(*d/ε*)^{d}−1 vertices, and the algorithm gives an approximation with*c(5**d*^{3}ln(1/ε)) vertices. The algorithms apply ideas previously used for small-dimensional linear programming.## Keywords

Convex Hull Range Query Query Time Successful Iteration Expected Time
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1993