On Computing the Vertex Centroid of a Polyhedron

TL;DR

This paper investigates the computational complexity of calculating the vertex centroid of a polyhedron, proving it is #P-hard, but also showing that approximation is feasible under certain conditions, with implications for related problems.

Contribution

It establishes the #P-hardness of computing the vertex centroid and halfspace membership, and demonstrates that approximation is possible with an oracle for vertex counting, linking it to other computational problems.

Findings

Computing the vertex centroid of an -polytope is P-hard.

Deciding if the centroid lies in a halfspace is P-hard.

Approximation of the centroid is P-easy given a vertex counting oracle.

Abstract

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 $\delta>0$.