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 open whether the Hirsch conjecture is true for general $m{\times}n$--transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that 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 $m{\times}n$--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 $3{\times}n$--transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for $2{\times}n$--transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter. 
Steffen Borgwardt, Jes\'us A. De Loera, Elisabeth Finhold, Jacob   Miller
math.CO math.OC