A subexponential algorithm for abstract optimization problems

An abstract optimization problem (AOP) is a triple (H,<,φ) where H is a finite set, < a linear order on 2^H and φ an oracle that, for given F⊆G⊆H, determines whether F=min(2 ^G), and if not, returns a smaller set. To solve the problem means to find min(2^H). The author presents a randomized algorithm that solves any AOP with an expected number of O(e^O(√|H|)) oracle calls. In contrast, any deterministic algorithm needs to make 2^|H|-1 oracle calls in the worst case. The algorithm is applied to the problem of finding the minimum distance of two polyhedra in d-space, which gives the first subexponential bound in d for this problem. Another application is the computation of the smallest ball containing n points in d-space; the previous bounds for this problem were also exponential in d