 
Summary: ESTIMATING THE NUMBER OF VERTICES OF A
POLYHEDRON
David Avis*
Luc Devroye
School of Computer Science
McGill University
3480 University
Montr’eal, Qu’ebec, Canada
H3A 2A7
Dedicated to Godfried Toussaint, a convex hull pioneer
ABSTRACT
Given a polyhedron P by a list of inequalities we develop unbiased esti
mates of the number of vertices and bases of P. The estimates are based on
applying tree estimation methods to the reverse search technique. The time to
generate an unbiased estimate is essentially bounded by the time taken to solve a
linear program on P with the simplex method. Computational experience is
reported. The method can be applied to estimate the output size of other enumer
ation problems solvable by reverse search.
1. Introduction
Let A be an m by n matrix with m ³ n and let b an nvector. The convex polyhedron P is
