Combinatorics and Geometry of Transportation Polytopes: An Update

TL;DR

This paper surveys recent advances in the combinatorics and geometry of transportation polytopes, highlighting new results on their structure, properties, and unresolved questions from the past thirty years.

Contribution

It provides an updated overview of the field, including recent unpublished findings on the diameter of transportation polytope graphs and revisiting longstanding open problems.

Findings

Recent unpublished results on graph diameters of transportation polytopes

Updated understanding of combinatorial and geometric properties

Revisiting open questions from 1984 and 1986 surveys

Abstract

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.