# Vertex enumeration problem

In mathematics, the **vertex enumeration problem** for a polytope, a polyhedral cell complex, a hyperplane arrangement, or some other object of discrete geometry, is the problem of determination of the object's vertices given some formal representation of the object. A classical example is the problem of enumeration of the vertices of a convex polytope specified by a set of linear inequalities:[1]

where *A* is an *m*×*n* matrix, *x* is an *n*×1 column vector of variables, and *b* is an *m*×1 column vector of constants.

## Computational complexity

The computational complexity of the problem is a subject of research in computer science. For unbounded polyhedra, the problem is known to be NP-hard, more precisely, there is no algorithm that runs in polynomial time in the combined input-output size, unless P=NP [2].

A 1992 article by David Avis and Komei Fukuda[3] presents an algorithm which finds the *v* vertices of a polytope defined by a nondegenerate system of *n* inequalities in *d* dimensions (or, dually, the *v* facets of the convex hull of *n* points in *d* dimensions, where each facet contains exactly *d* given points) in time O(*ndv*) and space O(*nd*). The *v* vertices in a simple arrangement of *n* hyperplanes in *d* dimensions can be found in O(*n*^{2}*dv*) time and O(*nd*) space complexity. The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids.

## Notes

- Eric W. Weisstein
*CRC Concise Encyclopedia of Mathematics,*2002, ISBN 1-58488-347-2, p. 3154, article "vertex enumeration" - Leonid Khachiyan; Endre Boros; Konrad Borys; Khaled Elbassioni; Vladimir Gurvich (March 2008). "Generating All Vertices of a Polyhedron Is Hard".
*Discrete and Computational Geometry*.**39**(1–3): 174–190. doi:10.1007/s00454-008-9050-5. - David Avis; Komei Fukuda (December 1992). "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra".
*Discrete and Computational Geometry*.**8**(1): 295–313. doi:10.1007/BF02293050.

## References

- Avis, David; Fukuda, Komei (December 1992). "A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra".
*Discrete and Computational Geometry*.**8**(1): 295–313. doi:10.1007/BF02293050. MR 1174359.