Abstract

An Abstract Optimization Problem AOP is a triple H where H is a finite set a total order on H and an oracle that for given F G H either reports that F min fF Gg or returns a set F G with F F To solve the problem means to find the minimum set in H We present a randomized algorithm that solves any AOP with an expected number of at most e p nO p n ln n oracle calls n jHj In contrast any deterministic algorithm needs to make n oracle calls in the worst case The algorithm is applied to the problem of finding the minimum distance between two n vertex or nfacet polyhedra in dspace and to the computation of the smallest ball containing n points in dspace for both problems we give the first subexponential bounds in d.