On the number of solutions generated by the dual simplex method

Abstract

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the dual simplex method with the most negative pivoting rule for LP. The bound is comparable with the bound given by Kitahara and Mizuno (2010) for the primal simplex method. We apply the result to the maximum flow problem and get a strong polynomial bound.

A primal-simplex based Tardos’ algorithm

“…He observed that this approach is strongly polynomial if A is totally unimodular and the auxiliary problems are non-degenerate; that is, the basic variables are strictly positive for every basic feasible solution. The strong polynomiality is a consequence of Kitahara and Mizuno [3,4] results which extend in part Ye's result [8] for Markov decision problems and bounds the number of distinct basic feasible solutions generated by the simplex method.…”

“…Therefore, since x ′′ is an optimal solution of P ′′ , we observe that x ′′ − x * ∞ < n ′ by Theorem 2 and thus, x * i > 0 for i ∈ J. Finally, we show the strong polynomiality of the proposed algorithm using Kitahara and Mizuno [3,4] results showing that the number of different basic feasible solutions generated by the primal simplex method with the most negative pivoting rule -Dantzig's rule -or the best improvement pivoting rule is bounded by:…”