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:

$Ax \leq b$

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.

A 1992 article by David Avis and Komei Fukuda 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(n²dv) time and O(nd) space complexity. The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids.