Random Walks on Polytopes of Constant Corank

TL;DR

This paper establishes a new lower bound on the expected number of pivoting steps in certain geometric algorithms, showing they can require logarithmic powers of the number of points, extending previous bounds to higher dimensions.

Contribution

It provides the first known lower bounds for pivoting processes in higher-dimensional polytopes of constant corank, applicable to linear programming and grid walks.

Findings

Lower bound of (\u2113^r n) steps for pivoting processes

Applicable to simplex algorithms with n constraints and n-r variables

Valid for directed random walks on grid polytopes of corank r

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 $n \to \infty$. The only cases for which the bound was known previously were for $r \le 3$. 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.