On Computing the Vertex Centroid of a Polyhedron

Let P be an H-polytope in R with vertex set V . The vertex centroid is defined as the average of the vertices in V . We prove that computing the vertex centroid of an H-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for H-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an ǫ distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an H-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to any “sufficiently” non-trivial (for example constant) distance, can be used to construct a fully polynomial approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of d 1 2 −δ for any fixed constant δ > 0.