On the Vector Linear Solvability of Networks and Discrete Polymatroids

We consider the vector linear solvability of networks over a field Fq. It is well known that a scalar linear solution over Fq exists for a network if and only if the network is matroidal with respect to a matroid representable over Fq. A discrete polymatroid is the multi-set analogue of a matroid. In this paper, a discrete polymatroidal network is defined and it is shown that a vector linear solution over a field Fq exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over Fq. An algorithm to construct networks starting from a discrete polymatroid is provided. Every representation over Fq for the discrete polymatroid, results in a vector linear solution over Fq 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 Fq.