How Good are Convex Hull Algorithms?

A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj+ jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability to deal well with \degeneracies," or, inability to control the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly dwarfed polytopes cause algorithms with bad intermed...

• Type: Conference
• Year: 1995
• Where: COMPGEOM
• Authors: David Avis, David Bremner
• Keywords: COMPGEOM 1995, Discrete Geometry, Enumeration Problem, Polytopes Cause Algorithms, Vertex Enumeration Problem