Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method

Abstract

This dissertation investigates the geometric combinatorics of convex polytopes and connections to the behavior of the simplex method for linear programming. We focus our attention on transportation polytopes, which are sets of all tables of non-negative real numbers satisfying certain summation conditions. Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure. First, we give new results on the diameters of certain classes of transportation polytopes and their relation to the Hirsch Conjecture, which asserts that the diameter of every $d$-dimensional convex polytope with $n$ facets is bounded above by $n-d$. In particular, we prove a new quadratic upper bound on the diameter of $3$-way axial transportation polytopes defined by $1$-marginals. We also show that the Hirsch Conjecture holds for $p \times 2$ classical transportation polytopes, but that there are infinitely-many Hirsch-sharp classical transportation polytopes. Second, we present new results on subpolytopes of transportation polytopes. We investigate, for example, a non-regular triangulation of a subpolytope of the fourth Birkhoff polytope $B_4$. This implies the existence of non-regular triangulations of all Birkhoff polytopes $B_n$ for $n \geq 4$. We also study certain classes of network flow polytopes and prove new linear upper bounds for their diameters.