On Supmodular Matrices

TL;DR

This paper investigates the permutability of matrices to become supermodular, revealing that small-dimensional matrices can always be permuted to be supermodular, unlike higher-dimensional matrices.

Contribution

It establishes the existence of universal permutations for small matrices and proves their non-existence in higher dimensions, raising open questions about permutability.

Findings

Small matrices are permutable to supermodular matrices by a universal permutation.

Higher-dimensional matrices do not admit a universal permutation for supermodularity.

Open questions remain about the dimensions where all matrices are permutable.

Abstract

We consider the problem of determining which matrices are permutable to be supmodular. We show that for small dimensions any matrix is permutable by a universal permutation or by a pair of permutations, while for higher dimensions no universal permutation exists. We raise several questions including to determine the dimensions in which every matrix is permutable.