Algorithms for polytope covering and approximation

Abstract

This paper gives an algorithm for 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(5dln c), for c large enough. The algorithm requires O(c ² n ^1+δ) expected time. More exactly, the time bound is

$$O(cn^{1 + \delta } + c(nc)^{1/(1 + \gamma /(1 + \delta ))} )$$

, where γγ1/[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.