How Good are Convex Hull Algorithms? \Lambda David Avis David Bremner y 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 Collections: Computer Technologies and Information Sciences