Two New Bounds for the Random‐Edge Simplex‐Algorithm

We prove that the RANDOM-EDGE simplex-algorithm requires an expected number of at most $13n/\sqrt{d}$ 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 $2^d$ on the performance of RANDOM-EDGE can asymptotically be improved by the factor $1/d^{(1-\varepsilon)\log d}$ for every $\varepsilon>0$.