the double-pivot algorithm solves these special LP problems in a strongly polynomial time.

The tightness of the bounds in the two theorems can be seen from the following problem provided in [@km11]: $$\begin{aligned}
\begin{array}{cl}
\min & -\sum_{i=1}^m   x_i \\
\mbox{subject to} & 
x_1 + x_{m+1} = 1, \\
& 2 \sum_{i=1}^{k-1} x_i +x_k + x_{m+k} = 2^k-1
 \hspace{0.1in} k=2, \ldots, m, 
  \\
& x_i \ge 0 \hspace{0.1in} i=1, \ldots, 2m.
\end{array}
\label{3rdProblem}\end{aligned}$$ Assuming that the initial point is taken as $\x^0 =[0,\ldots,0,1,\ldots,1]$ (there are $m$ zeros and $n-m$ ones) and Dantzig’s rule is used, for this problem, Kitathra and Mizuno showed [@km11] that the bound of Theorem \[thm2\] is reduced to $\lceil (2m \log 2)2^m \rceil$, while the actual iteration number is $2^m-1$. The estimated bound is reasonably tight. We show that the bound of Theorem \[mainThe\] is much tighter than the one of Theorem \[thm2\]. For this problem, it is easy to see that the first $m$ variables of the optimal solution are $[x_1^*,\ldots,x_m^*]=[ 0, \ldots, 0,2^m-1]$ with optimal objective function $-(2^m-1)$ and the objective function at initial $\x^0$ is zero. Therefore, we have $\c^{\Tr}\x^0-z^*=2^m-1$. Since every component of any basic feasible solution is a positive integer (because every 