文摘
Let be an -polytope in with vertex set . The vertex centroid is defined as the average of the vertices in . We first prove that computing the vertex centroid of an-polytope, or even just checking whether it lies in a given halfspace, is #P-hard. We also consider the problem of approximating the vertex centroid by finding a point within an distance from it and prove this problem to be #P-easy in the sense that it can be solved efficiently using an oracle for some #P-complete problem. In particular, we show that given an oracle for counting the number of vertices of an -polytope, one can approximate the vertex centroid in polynomial time. Counting the number of vertices of a polytope defined as the intersection of halfspaces is known to be #P-complete. 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-time 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 cannot be approximated to a distance of for any fixed constant unless .
