The hierarchy of circuit diameters and transportation polytopes

The hierarchy of circuit diameters and transportation polytopes The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still unknown whether the Hirsch conjecture is true for general m×n-transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. This hierarchy provides some interesting lower bounds for the usual graph diameter.This paper has three contributions: First, we compare the hierarchy of diameters for the m×n-transportation polytopes. We show that the Hirsch conjecture bound of m+n−1 is actually valid in most of these diameter notions. Second, we prove that for 3×n-transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2×n-transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Applied Mathematics Elsevier

The hierarchy of circuit diameters and transportation polytopes

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Publisher
North-Holland
Copyright
Copyright © 2015 Elsevier B.V.
ISSN
0166-218X
D.O.I.
10.1016/j.dam.2015.10.017
Publisher site
See Article on Publisher Site

Abstract

The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still unknown whether the Hirsch conjecture is true for general m×n-transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. This hierarchy provides some interesting lower bounds for the usual graph diameter.This paper has three contributions: First, we compare the hierarchy of diameters for the m×n-transportation polytopes. We show that the Hirsch conjecture bound of m+n−1 is actually valid in most of these diameter notions. Second, we prove that for 3×n-transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2×n-transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter.

Journal

Discrete Applied MathematicsElsevier

Published: May 11, 2018

References

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