On Computing the Vertex Centroid of a Polyhedron

Let $\mathcal{P}$ be an $\mathcal{H}$-polytope in $\mathbb{R}^d$ 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 $\mathcal{H}$-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 $\mathcal{H}$-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an $\epsilon$ 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 $\mathcal{H}$-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to \emph{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 $d^{{1/2}-\delta}$ for any fixed constant $\delta>0$.