The average number of pivot steps of the simplex-algorithm based on a generalized rotation-symmetry-model

Abstract

This paper deals with the average-case-analysis of the number of pivot steps required by the simplex method. It generalizes results of Borgwardt (who worked under the assumpution of the rotation-symmetry-model) for the shadow-vertex-algorithm to so-called cylindric distributions. Simultaneously it allows to analyze an extended dimension-by-dimension-algorithm, which solves linear programing problems with arbitrary capacity bounds $b$ in the restrictions $Ax\le b$, whereas the model used by Borgwardt required strictly positive right hand sides $b$. These extensions are achieved by solving a problem of stochastic geometry closely related to famous results of Renyi and Sulanke, namely: assume that $a_1,\ldots ,a_m$ are uniformly distributed in a cylinder. How many facets of ${{\mathrm{conv}}}(a_1,\ldots ,a_m,0)$ will be intersected by a two-dimensional shadow plane along the axis of the cylinder? The consequence of these investigations is that the upper bounds of Borgwardt (under his original model) still apply when we accept distributions with arbitrary right hand sides.