Vertex enumeration problem: Difference between revisions

Revision as of 14:08, 16 February 2019

Latest revision as of 18:49, 6 August 2022

(6 intermediate revisions by 5 users not shown)

Line 2:

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

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 inverse ([[Duality (mathematics)|dual]]) problem of finding the bounding inequalities given the vertices is called ''[[facet enumeration]]'' (see [[convex hull algorithms]]).

==Computational complexity==

The [[Computational complexity theory|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 .<ref>{{cite journal|author1=Leonid Khachiyan |author2=Endre Boros |author3=Konrad Borys |author4=Khaled Elbassioni |author5=Vladimir Gurvich |title=Generating All Vertices of a Polyhedron Is Hard |journal=[[Discrete and Computational Geometry]] |volume=39 |number=1-31–3 |date=March 2008 |pages=174–190 |doi= 10.1007/s00454-008-9050-5|doi-access=free }}</ref>.

A 1992 article by [[David Avis]] and [[Komei Fukuda]]<ref>{{cite journal|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=295–313 |doi=10.1007/BF02293050|doi-access=free }}</ref> presents ana [[reverse-search 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==

{{reflist}}

==References==

* {{cite journal|first1=David|last1=Avis|first2=Komei|last2=Fukuda|authorlink2=Komei Fukuda|authorlink1=David Avis|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=295–313

|doi=10.1007/BF02293050|mr=1174359|refdoi-access=harvfree}}

[[Category:Geometric algorithms]]