Random walks on polytopes of constant corank

Abstract

We show that the pivoting process associated with one line and n points in r-dimensional space may need Omega(log^r n) steps in expectation as n -> infty. The only cases for which the bound was known previously were for r <= 3. Our lower bound is also valid for the expected number of pivoting steps in the following applications: (1) The Random-Edge simplex algorithm on linear programs with n constraints in d = n-r variables; and (2) the directed random walk on a grid polytope of corank r with n facets.