Small Degenerate Simplices Can Be Bad for Simplex Methods

Abstract

We show that the simplex method with Dantzig's pivoting rule may require an exponential number of iterations over two highly degenerate instances. The feasible region of the first instance is a full dimensional simplex, and a single point for the second one. In addition, the entries of the constraint matrix, the right-hand-side vector, and the cost vector are {0, 1, 2}-valued. Those instances, with few vertices and small input data length, illustrate the impact of degeneracy on simplex methods.