Perturbation of transportation polytopes

TL;DR

This paper introduces a perturbation method to simplify the computation of multivariate generating functions for transportation polytopes, enabling explicit formulas and enumeration of combinatorial structures.

Contribution

It develops a universal perturbation technique for transportation polytopes and derives explicit MGFs and combinatorial formulas for their vertices and feasible cones.

Findings

Derived formulas for MGFs of central transportation polytopes

Reproduced known results for Birkhoff polytopes

Counted maximum vertices of transportation polytopes

Abstract

We describe a perturbation method that can be used to reduce the problem of finding the multivariate generating function (MGF) of a non-simple polytope to computing the MGF of simple polytopes. We then construct a perturbation that works for any transportation polytope. We apply this perturbation to the family of central transportation polytopes of order kn x n, and obtain formulas for the MGFs of the feasible cone of each vertex of the polytope and the MGF of the polytope. The formulas we obtain are enumerated by combinatorial objects. A special case of the formulas recovers the results on Birkhoff polytopes given by the author and De Loera and Yoshida. We also recover the formula for the number of maximum vertices of transportation polytopes of order kn x n.