1Dept. of Industrial Engineering and Operations Research, Univ. of California, Berkeley 2Department of Mathematics, Univ. of California, Davis 3Department of Mathematics, Seattle University
The electronic journal of combinatorics, Tome 28 (2021) no. 4
Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.
Ilan Adler
1
;
Jesús A. De Loera
2
;
Steven Klee
3
;
Zhenyang Zhang
2
1
Dept. of Industrial Engineering and Operations Research, Univ. of California, Berkeley
2
Department of Mathematics, Univ. of California, Davis
3
Department of Mathematics, Seattle University
@article{10_37236_9653,
author = {Ilan Adler and Jes\'us A. De Loera and Steven Klee and Zhenyang Zhang},
title = {Diameters of cocircuit graphs of oriented matroids: an update},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {4},
doi = {10.37236/9653},
zbl = {1486.52057},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9653/}
}
TY - JOUR
AU - Ilan Adler
AU - Jesús A. De Loera
AU - Steven Klee
AU - Zhenyang Zhang
TI - Diameters of cocircuit graphs of oriented matroids: an update
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/9653/
DO - 10.37236/9653
ID - 10_37236_9653
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%0 Journal Article
%A Ilan Adler
%A Jesús A. De Loera
%A Steven Klee
%A Zhenyang Zhang
%T Diameters of cocircuit graphs of oriented matroids: an update
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/9653/
%R 10.37236/9653
%F 10_37236_9653
Ilan Adler; Jesús A. De Loera; Steven Klee; Zhenyang Zhang. Diameters of cocircuit graphs of oriented matroids: an update. The electronic journal of combinatorics, Tome 28 (2021) no. 4. doi: 10.37236/9653
