Klee-Minty's LP and Upper Bounds for Dantzig's Simplex Method

TL;DR

This paper explores the relationship between the ratio of positive element values in LP solutions and the number of Dantzig's simplex iterations, providing insights into upper bounds and their limitations.

Contribution

It establishes a direct link between the ratio of positive element values and iteration count, and offers improved upper bounds for the simplex method.

Findings

The ratio equals the number of iterations in a specific LP variant.

Upper bounds depend critically on the ratio of element values.

Improved bounds are proposed based on this relationship.

Abstract

Kitahara and Mizuno (2010) get two upper bounds for the number of different basic feasible solutions generated by Dantzig's simplex method. The size of the bounds highly depends on the ratio between the maximum and minimum values of all the positive elements of basic feasible solutions. In this paper, we show some relations between the ratio and the number of iterations by using an example of LP, which is a simple variant of Klee-Minty's LP. We see that the ratio for the variant is equal to the number of iterations by Dantzig's simplex method for solving it. This implies that it is impossible to get a better upper bound than the ratio. We also give improved results of the upper bounds.