Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids

TL;DR

This paper establishes a complete characterization of scalar-linear solvability in matroidal networks, showing it is equivalent to being associated with a representable matroid over a finite field, and provides methods to generate such networks.

Contribution

It proves the converse of a key theorem, linking scalar-linear solvability to representable matroids, and demonstrates solvability for networks from uniform and graphic matroids.

Findings

Scalar-linear solvability is equivalent to being associated with a representable matroid.

Provides a method to generate scalar-linearly solvable networks.

Demonstrates solvability for networks from uniform and graphic matroids.

Abstract

We study matroidal networks introduced by Dougherty et al. We prove the converse of the following theorem: If a network is scalar-linearly solvable over some finite field, then the network is a matroidal network associated with a representable matroid over a finite field. It follows that a network is scalar-linearly solvable if and only if the network is a matroidal network associated with a representable matroid over a finite field. We note that this result combined with the construction method due to Dougherty et al. gives a method for generating scalar-linearly solvable networks. Using the converse implicitly, we demonstrate scalar-linear solvability of two classes of matroidal networks: networks constructed from uniform matroids and those constructed from graphic matroids.