Tropical Determinant of Integer Doubly-Stochastic Matrices

TL;DR

This paper establishes the exact maximum and minimum values of a modified tropical determinant over integer points in scaled Birkhoff polytopes, advancing understanding of combinatorial matrix properties.

Contribution

It provides the first precise bounds on the tropical determinant for integer doubly-stochastic matrices within scaled Birkhoff polytopes.

Findings

Sharp upper bound on tropical determinant over D(m,n)

Sharp lower bound on modified tropical determinant over D(m,n)

Characterization of extremal matrices for these bounds

Abstract

Let D(m,n) be the set of all the integer points in the m-dilate of the Birkhoff polytope of doubly-stochastic n by n matrices. In this paper we find the sharp upper bound on the tropical determinant over the set D(m,n). We define a version of the tropical determinant where the maximum over all the transversals in a matrix is replaced with the minimum and then find the sharp lower bound on thus defined tropical determinant over D(m,n).