Random Walks on Polytopes of Constant Corank

02/26/2018

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by   Malte Milatz, et al.

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We show that the pivoting process associated with one line and n points in r-dimensional space may need Ω(^r n) steps in expectation as n →∞. 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.