An Upper Bound for the Number of Solutions (Vertices) Generated by the Simplex Method
Shinji Mizuno, Tokyo Institute of Technology
We give an upper bound for the number of different basic feasible solutions generated by the simplex method for a linear programming problem which has 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 LPs. The result includes strongly polynomiality of the simplex method for Markov Decision Problem by Ye.
Joint work with Tomonari Kitahara