Vertex enumeration problem: Difference between revisions

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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 [[vertex (geometry)|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]]:<ref>[[Eric W. Weisstein]] ''CRC Concise Encyclopedia of Mathematics,'' 2002, {{isbn|1-58488-347-2}}, p. &nbsp;3154, article "vertex enumeration"</ref>

:<math>Ax \leq b</math>

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The [[Computational complexity theory|computational complexity]] of the problem is a subject of research in [[computer science]].

A 1992 article by [[David Avis]] and Komei Fukuda<ref>{{cite journal|url=http://www.springerlink.com/content/m7440v7p3440757u/ |author1=David Avis |author2=Komei Fukuda |title=A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra |journal=[[Discrete and Computational Geometry]] |volume=8 |number=1 |date=December 1992 |pages=295-313295–313 |doi=10.1007/BF02293050}}</ref> 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'' [[facet]]s of the [[convex hull]] of ''n'' points in ''d'' dimensions, where each facet contains exactly ''d'' given points) in time [[Big Oh notation|O]](''ndv'') and [[space complexity|space]] O(''nd''). The ''v'' vertices in a simple arrangement of ''n'' [[hyperplane]]s 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==