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

Loading next page...
 
/lp/elsevier/the-hierarchy-of-circuit-diameters-and-transportation-polytopes-05dcu5CReK
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

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off