The average number of pivot steps of the simplex-algorithm based on a generalized rotation-symmetry-model
<Para ID="Par1">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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$b$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>b</mi> </math> </EquationSource> </InlineEquation> in the restrictions <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Ax\le b$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>A</mi> <mi>x</mi> <mo>≤</mo> <mi>b</mi> </mrow> </math> </EquationSource> </InlineEquation>, whereas the model used by Borgwardt required strictly positive right hand sides <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$b$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi>b</mi> </math> </EquationSource> </InlineEquation>. These extensions are achieved by solving a problem of stochastic geometry closely related to famous results of Renyi and Sulanke, namely: assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$a_1,\ldots ,a_m$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> </mrow> </math> </EquationSource> </InlineEquation> are uniformly distributed in a cylinder. How many facets of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${{\mathrm{conv}}}(a_1,\ldots ,a_m,0)$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="normal">conv</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>m</mi> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math> </EquationSource> </InlineEquation> 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. Copyright Springer-Verlag Berlin Heidelberg 2014
