On Computing the Vertex Centroid of a Polyhedron
| dc.creator | Elbassioni, Khaled | |
| dc.creator | Tiwary, Hans Raj | |
| dc.date | 2008-06-20 | |
| dc.date.accessioned | 2026-07-07T12:19:38Z | |
| dc.date.available | 2026-07-07T12:19:38Z | |
| dc.description | 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 $ε$ 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 $d^{{1/2}-δ}$ for any fixed constant $δ>0$. | |
| dc.identifier | https://arxiv.org/abs/0806.3456 | |
| dc.identifier | http://arxiv.org/abs/0806.3456 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/212827 | |
| dc.subject | Computational Geometry | |
| dc.title | On Computing the Vertex Centroid of a Polyhedron | |
| dc.type | text |