A bound for the number of different basic solutions generated by the simplex method

Abstract

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints, the number of variables, and the ratio between the minimum and the maximum values of all the positive elements of primal basic feasible solutions. When the primal problem is nondegenerate, it becomes a bound for the number of iterations. We show some basic results when it is applied to special linear programming problems. The results include strongly polynomiality of the simplex method for Markov Decision Problem by Ye [3] and utilize its analysis.

On the number of solutions generated by the dual simplex method

“…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.…”

“…The analysis in this section is similar to the one in Kitahara and Mizuno [3]. The next result corresponds to Theorem 3.1 in [3]. …”

“…Kitahara and Mizuno [3] give a bound for the number of different basic feasible solutions (BFSs) generated by the primal simplex method with Dantzig's rule. They extend the analysis by Ye [6] for the Markov Decision Problem to general LPs.…”

On the reduction of total‐cost and average‐cost MDPs to discounted MDPs

“…Remark Consider an LP with n constraints and m variables, where the positive elements of every basic feasible solution are bounded below by δ and bounded above by γ . By generalizing the analysis in Ye for discounted MDPs, it is proved in Kitahara and Mizuno , Theorem 3] that the simplex method with Dantzig's rule requires at most iterations to return an optimal solution. For the LP (46), δ = 1 and satisfy the hypotheses of this result.…”

“…For the LP (46), δ = 1 and satisfy the hypotheses of this result. Therefore, it follows from , Theorem 3] that an average‐cost optimal policy can be computed in strongly polynomial time when is fixed, by applying the simplex method with Dantzig's rule to the LP (46). However, , Theorem 3] does not imply an analogous statement for the LP (51) for unichain average‐cost MDPs.…”