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The Simplex Algorithm in Dimension Three (2004)
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@MISC{Kaibel04thesimplex,
author = {Volker Kaibel and Rafael Mechtel and Micha Sharir and Günter M. Ziegler},
title = {The Simplex Algorithm in Dimension Three },
year = {2004}
}
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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.
Keyphrases
simplex algorithm dimension three linear program dimension restriction complicated analysis deterministic history-dependent rule general dimension 3-dimensional simple polytopes worst-case behavior worst-possible behavior random-facet rule asymptotic behavior pivot rule non-polynomial instance random edge rule
