Random-Edge Is Slower Than Random-Facet on Abstract Cubes

Random-Edge and Random-Facet are two very natural randomized pivoting rules for the simplex algorithm. The behavior of Random-Facet is fairly well understood. It performs an expected sub-exponential number of pivoting steps on any linear program, or more generally, on any Acyclic Unique Sink Orientation (AUSO) of an arbitrary polytope, making it the fastest known pivoting rule for the simplex algorithm. The behavior of Random-Edge is much less understood. We show that in the AUSO setting, Random-Edge is slower than Random-Facet. To do that, we construct AUSOs of the n-dimensional hypercube on which Random-Edge performs an expected number of 2Ω( √ n logn) steps. This improves on a 2Ω( 3 √ n) lower bound of Matoušek and Szabó (FOCS 2004). As Random-Facet performs an expected number of 2O( √ n) steps on any n-dimensional AUSO, this established our result. Improving our 2Ω( √ n logn) lower bound seems to require radically new techniques. 1998 ACM Subject Classification G.1.6 Optimization Linear Programming; G.2.1 Combinatorics Combinatorial Algorithms