Tropical determinant on transportation polytope

TL;DR

This paper establishes the exact lower and upper bounds for the tropical determinant on a specific set of integer matrices within transportation polytopes, linking a complex piecewise-linear problem to a simpler quadratic optimization.

Contribution

It provides the first sharp bounds for the tropical determinant on transportation polytope integer points, connecting high-dimensional linear programming to a 2D quadratic problem.

Findings

Sharp lower bound on tropical determinant established

Sharp upper bound on modified tropical determinant computed

Reduction of a high-dimensional problem to a 2D quadratic optimization

Abstract

Let ${\mathcal D}^{k,l}(m,n)$ be the set of all the integer points in the transportation polytope of $kn\times ln$ matrices with row sums $lm$ and column sums $km$. In this paper we find the sharp lower bound on the tropical determinant over the set ${\mathcal D}^{k,l}(m,n)$. This integer piecewise-linear programming problem in arbitrary dimension turns out to be equivalent to an integer non-linear (in fact, quadratic) optimization problem in dimension two. We also compute the sharp upper bound on a modification of the tropical determinant, where the maximum over all the transversals in a matrix is replaced with the minimum.