The Random-Facet simplex algorithm on combinatorial cubes

The Random-Facet algorithm is a randomized variant of the simplex method which is known to solve any linear program with n variables and m constraints using an expected number of pivot steps which is subexponential in both n and m. This is the theoretically fastest simplex algorithm known to date if m ≈ n; it provably beats most of the classical deterministic variants which require exp(Ω(n)) pivot steps in the worst case. Random-Facet has independently been discovered and analyzed ten years ago by Kalai as a variant of the primal simplex method, and by Matoušek, Sharir and Welzl in a dual form. The essential ideas and results connected to Random-Facet can be presented in a particularly simple and instructive way for the case of linear programs over combinatorial n-cubes. I derive an explicit upper bound of n