On the Vector Linear Solvability of Networks and Discrete Polymatroids

TL;DR

This paper establishes a connection between the vector linear solvability of networks and discrete polymatroids, extending the matroidal framework to multi-set analogues and providing algorithms for network construction.

Contribution

It introduces the concept of discrete polymatroidal networks and proves their equivalence to vector linear solvability over finite fields, along with a construction algorithm.

Findings

Networks can be constructed from discrete polymatroids.

Vector linear solutions exist for these networks.

Some networks admit vector solutions but not scalar solutions.

Abstract

We consider the vector linear solvability of networks over a field $\mathbb{F}_q.$ It is well known that a scalar linear solution over $\mathbb{F}_q$ exists for a network if and only if the network is \textit{matroidal} with respect to a \textit{matroid} representable over $\mathbb{F}_q.$ A \textit{discrete polymatroid} is the multi-set analogue of a matroid. In this paper, a \textit{discrete polymatroidal} network is defined and it is shown that a vector linear solution over a field $\mathbb{F}_q$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An algorithm to construct networks starting from a discrete polymatroid is provided. Every representation over $\mathbb{F}_q$ for the discrete polymatroid, results in a vector linear solution over $\mathbb{F}_q$ for the constructed network. Examples which illustrate the construction algorithm are provided, in which the resulting networks admit vector linear solution but no scalar linear solution over $\mathbb{F}_q.$