 
Summary: How Good are Convex Hull Algorithms? \Lambda
David Avis David Bremner y
January, 1995
Abstract
A convex polytope is the bounded intersection of finite set H of halfspaces. A classic theorem
of convexity theory is that every convex polyhedron can be expressed as the convex hull of its
set V of vertices. There are three closely related computational problems related to the two
descriptions of a polytope. The vertex enumeration problem is to compute V from H. The
convex hull problem it to compute H from V. The polytope verification problem is to decide
whether a given vertex description and halfspace description define the same polytope. The
first two problems are essentially equivalent under point/hyperplane duality. It is an open
problem whether any of these problems can be solved in time polynomial in jHj + jVj. In this
paper we describe hard polytopes for convex hull algorithms based on pivoting , those based on
triangulation, and for some insertion algorithms.
1 Introduction
Although the simplex method had long been regarded as a practical and efficient algorithm for
linear programming, it was not until the seminal 1972 paper of Klee and Minty [13] that it was
demonstrated to be an exponential time algorithm. The authors described a class of hard polytopes
that cause the simplex method to make an exponential number of pivots using the greatest cost
coefficient pivoting rule. Subsequent papers gave similar results for other pivot rules, and the
