The Simplex Algorithm in Dimension Three
Abstract
We investigate the worst-case behavior of the simplex algorithm on linear programs with three 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 without dimension restriction is known to be subexponential, as well as Zadeh's deterministic history dependent rule, for which no non-polynomial instances in general dimensions have been found so far.
Cite
@article{arxiv.math/0309351,
title = {The Simplex Algorithm in Dimension Three},
author = {Volker Kaibel and Rafael Mechtel and Micha Sharir and Guenter M. Ziegler},
journal= {arXiv preprint arXiv:math/0309351},
year = {2007}
}
Comments
24 pages, to appear in: SIAM J. Comp.; the paper comprises the contents of our paper 'The Random Edge Rule on Three-Dimensional Linear Programs' (math.CO/0301076)