Algorithms for polytope covering and approximation

  • Kenneth L. Clarkson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 709)


This paper gives an algorithm for polytope covering: let L and U be sets of points in Rd, 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(5dln c), for c large enough. The algorithm requires O(c2n1+δ) expected time. More exactly, the time bound is
$$O(cn^{1 + \delta } + c(nc)^{1/(1 + \gamma /(1 + \delta ))} )$$
, where γγ1/[d/2]. The previous best bounds were cO(log n) cover size in O(nd) 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(5d3 ln(1/ε)) vertices. The algorithms apply ideas previously used for small-dimensional linear programming.


Convex Hull Range Query Query Time Successful Iteration Expected Time 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Kenneth L. Clarkson
    • 1
  1. 1.AT&T Bell LaboratoriesMurray Hill

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