A bound for the number of different basic solutions generated by the simplex method - Kitahara, Tomonari; Mizuno, Shinji (Mathematical programming, 2013)
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 (LP) 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 problem is primal nondegenerate, it becomes a bound for the number of iterations. The result includes strong polynomiality for Markov Decision Problem by Ye ( http://www.stanford.edu/~yyye/simplexmdp1.pdf , 2010 ) and utilize its analysis. We also apply our result to an LP whose constraint matrix is totally unimodular and a constant vector b of constraints is integral.