Average-Case Analysis of the Double Description Method and the Beneath-Beyond Algorithm

Abstract

This paper deals with the average computational effort for calculating all vertices of a polyhedron described by m inequalities in an n-dimensional space, when we apply the so-called "Double Description Method" (from a dual point of view, i.e. for finding all facets of the convex hull of m given points, this is equivalent to application of the "Beneath-Beyond Algorithm"). Both are incremental algorithms, i.e. they develop the information about the polyhedron stepwise by taking the inequalities/points successively into regard. The average-case analysis is done with respect to the Rotation-Symmetry Model, which is well known from the corresponding analysis of the Simplex Method for linear programming. In this model degenerate problems occur with probability 0. So the (finite) effort to solve those problems has no impact on the expected effort in our model. All the derived results and complexities apply equivalently to both algorithms and to the corresponding primal and dual problems.