We investigate the worst-case behavior of the simplex algorithm on linear programs
with 3 variables, that is, on 3-dimensional simple polytopes. Among the
pivot rules that we consider, the “random edge” rule yields the best asymptotic
behavior as well as the most complicated analysis. All other rules turn out to be
much easier to study, but also produce worse results: Most of them show essentially
worst-possible behavior; this includes both Kalai’s “random-facet” rule, which is
known to be subexponential without dimension restriction, as well as Zadeh’s deterministic
history-dependent rule, for which no non-polynomial instances in general
dimensions have been found so far.
Revlex-Initial 0/1-Polytopes
(2005)