Algorithms for polytope covering and approximation

Abstract

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(5d ln c), for c large enough. The algorithm requires O(c2n1+δ) expected time.1 More exactly, the time bound is O(cn1+δ + c(nc)1/(1+γ/(1 +δ))), where γ≡1/[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 e 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.