On the Number of Solutions Generated by the Simplex Method for LP

We obtain upper bounds for the number of distinct solutions generated by the simplex method for linear programming (LP). One of the upper bounds is polynomial in the number of variables, the number of constraints, and the ratio of the maximum to the minimum positive components in all the basic feasible solutions. We show that they are good upper bounds for some special LP problems including those on 0-1 polytopes, those with totally unimodular matrices, and the Markov decision problems. We also show that the upper bounds are almost tight by using an LP instance on a 0-1 polytope and a simple variant of the Klee-Minty example.