2003/09/22 by Volker Kaibel, Kaibel, Volker, Rafael Mechtel +5
Mathematics · #52B10 #68W20 #90C05 #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Control (math.OC) #math.CO #math.OC #msc:52B10 #msc:68W20 #msc:90C05
paper · pdf · doi:10.48550/arxiv.math/0309351
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)
arxiv created 2004/08/20 · arxiv updated 2009/12/01
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.