2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/112656The simplex algorithm using the random edge pivot-rule on any realization of a dual cyclic 4-polytope with n facets does not take more than O(n) pivot-steps. This even holds for general abstract objective functions (AOF) / acyclic unique sink orientations (AUSO). The methods can be used to show analogous results for products of two polygons. In contrast, we show that the random facet pivot-rule is slow on dual cyclic 4-polytopes, i.e. there are AUSOs on which random facet takes at least Ω(n^2) steps.31 Pages, 11 figuresCombinatoricsOptimization and Control90C05 (Primary); 52B12, 68W20 (Secondary)text