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

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.