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arXiv · 1410.3397

Tropical determinant on transportation polytope

Sailaja Gajula·Ivan Soprunov·Jenya Soprunova
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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.

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Sailaja Gajula, Ivan Soprunov, Jenya Soprunova. 2014-10-13. Tropical determinant on transportation polytope. https://doi.org/10.1016/j.laa.2015.01.036

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