Long and winding central paths
Abstract
We disprove a continuous analogue of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with inequalities in dimension where the central path has a total curvature in . Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. The lower bound for the classical curvature is obtained by developing a combinatorial concept of a tropical angle.
Keywords
Cite
@article{arxiv.1405.4161,
title = {Long and winding central paths},
author = {Xavier Allamigeon and Pascal Benchimol and Stéphane Gaubert and Michael Joswig},
journal= {arXiv preprint arXiv:1405.4161},
year = {2017}
}
Comments
This paper is superseded by arXiv:1708.01544. 27 pages, 4 figures, 2 tables. v2: Major revision which includes: * a primal-dual description of the tropical central path, * a uniform bound on the distance between the classical and tropical central paths, * an improved analysis of the curvature relying on a notion of tropical angle