TY - JOUR
T1 - The simplex algorithm in dimension three
AU - Kaibel, Volker
AU - Mechtel, Rafael
AU - Sharir, Micha
AU - Ziegler, Günter M.
PY - 2005
Y1 - 2005
N2 - 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, and Zadeh's deterministic history-dependent rule, for which no nonpolynomial instances in general dimensions have been found so far.
AB - 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, and Zadeh's deterministic history-dependent rule, for which no nonpolynomial instances in general dimensions have been found so far.
KW - Linear programming
KW - Linearity coefficient
KW - Pivot rule
KW - Random edge
UR - http://www.scopus.com/inward/record.url?scp=18444404378&partnerID=8YFLogxK
U2 - 10.1137/S0097539703434978
DO - 10.1137/S0097539703434978
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AN - SCOPUS:18444404378
SN - 0097-5397
VL - 34
SP - 475
EP - 497
JO - SIAM Journal on Computing
JF - SIAM Journal on Computing
IS - 2
ER -