A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra

Abstract

We present a newpiv ot-based algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties:(a) No additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for non-degenerate inputs; (e) The algorithm is easy to efficiently parallelize.Forexample, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually,the v facets of the convex hull of n points in R d ,w here each facet contains exactly d givenp oints) in time O(ndv)a nd O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n 2 dv)t ime and O(nd)s pace complexity.T he algorithm is based on inverting finite pivotalgorithms for linear programming.-2-

Computation of elementary modes: a unifying framework and the new binary approach

“…The algorithms related to EM computation may be, in general, especially suited for computing extreme rays in such strongly degenerate systems, whereas other programs may be better suited for only weakly degenerate problems. For example, the software lrs [33] implements the so-called reverse search enumeration algorithm [37] that is polynomial for non-degenerate cases. Note that the new binary approach as introduced herein can easily be adapted for computing extreme rays of any pointed cone as given in eq.…”

On the Dynamic Problem of Computing Generators of a Polyhedral Cone

“…Задача построе-ния двойственного описания полиэдра является одной из центральных в теории систем линейных неравенств [1][2][3][4]. Известно множество алгоритмов решения данной задачи [5][6][7][8][9][10] и близких к ней задач, включая вопросы трудоемкости и организации параллельных вычислений [11][12][13][14]. Задача построения двойственного описания выпуклого полиэдра в ℚ d сводится к аналогичной задаче для полиэдрального конуса в ℚ d+1 (см., например, [4]).…”

An algorithm for constructing the skeleton graph of degenerate systems of linear inequalities

“…The most effective conversion methods available arise from combinatorial geometry [16–19]. However, when the half-space descriptions of the extreme points of the feasible polytope are degenerate a combinatorial explosion is produced that is the cause of stubborn difficulties for their enumeration [19–23].…”

“…The combinatorial triviality of regular extreme points does not present any difficulty to the simplex pivoting rules [16, 24, 25]. However, when pivoting around a σ -degenerate extreme point , the method faces up to potential bases in the active set , so the exhaustive search of the neighboring points of requires the examination of simplex tableaus and a factible pivote has to be looked for by testing the n entries of m − n − σ rows of every tableau.…”