Deformed Products and Maximal Shadows of Polytopes (1996)
| Venue: | ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223 |
| Citations: | 24 - 1 self |
BibTeX
@INPROCEEDINGS{Amenta96deformedproducts,
author = {Nina Amenta and Günter M. Ziegler},
title = {Deformed Products and Maximal Shadows of Polytopes},
booktitle = {ADVANCES IN DISCRETE AND COMPUTATIONAL GEOMETRY, AMER. MATH. SOC., PROVIDENCE, CONTEMPORARY MATHEMATICS 223},
year = {1996},
pages = {57--90},
publisher = {}
}
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Abstract
We present a construction of deformed products of polytopes that has as special cases all the known constructions of linear programs with "many pivots," starting with the famous Klee-Minty cubes from 1972. Thus we obtain sharp estimates for the following geometric quantities for d-dimensional simple polytopes with at most n facets: ffl the maximal number of vertices on an increasing path, ffl the maximal number of vertices on a "greedy" greatest increase path, and ffl the maximal number of vertices of a 2-dimensional projection. This, equivalently, provides good estimates for the worst-case behaviour of the simplex algorithm on linear programs with these parameters with the worst-possible, the greatest increase, and the shadow vertex pivot rules. The bounds on the maximal number of vertices on an increasing path or a greatest increase path unify and slightly improve a number of known results. One bound on the maximal number of vertices of a 2-dimensional projection is new: we show ...
