A note on the diameter of transportation polytopes with prescribed source degrees

TL;DR

This paper examines the diameters of specific classes of transportation polytopes with prescribed source degrees, building on prior bounds and contributing to understanding their combinatorial properties.

Contribution

It investigates the diameters of certain transportation polytopes, providing bounds and insights that relate to earlier work by Balinski and Rispoli.

Findings

Bounds on diameters for specific transportation polytope classes

Connections to previous results by Balinski and Rispoli

Refinements and generalizations of transportation polytope properties

Abstract

Brightwell, van den Heuvel and Stougie proved that the diameter of an $m \times n$ transportation polytope is at most $8(m+n-2)$, a factor of eight away from the Hirsch Conjecture. This bound was improved to $3(m+n-1)$ by Hurkens. We investigate diameters for certain classes of transportation polytopes. Note: After the completion of this note, we discovered that the class of transportation polytopes studied in this note was already considered in Michel L. Balinski. On two special classes of transportation polytopes. Math. Programming Stud., 1:43-58, 1974. Michel L. Balinski and Fred J. Rispoli. Signature classes of transportation polytopes. Mathematical Programming, 60(2, Ser. A):127-144, 1993. These papers contain both refinements of our results and generalizations to more general classes of transportation problems. In view of these papers, this note will not be submitted for publication.