Two New Bounds for the Random-Edge Simplex-Algorithm

We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/ √ d pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2 on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.