Randomized Simplex Algorithms on Klee-Minty Cubes

We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we develop combinatorial models for the Klee-Minty cubes [17] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the Klee-Minty cubes leads t o (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds conjectured in the literature. At the same time, we establish quadratic upper bounds for random pivots on the linear programs under investigation. This motivates the question whether some randomized pivot rules possibly have quadratic worst-case behavior on general linear programs.