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:
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.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 .
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 in time O and space O. The v vertices in a simple arrangement of n hyperplanes in d dimensions can be found in O time and O space complexity. The Avis–Fukuda algorithm adapted the criss-cross algorithm for oriented matroids.
