Scalar Solvability of Network Computation Problems and Representable Matroids

TL;DR

This paper establishes a connection between network computation problems and representable matroids, showing scalar linear solutions correspond to matroidal structures, and extends the framework to nonlinear network codes via FD-relations.

Contribution

It introduces a novel relation between network computation problems and representable matroids, and characterizes nonlinear network codes using FD-relations.

Findings

Scalar linear solutions exist iff the problem is matroidal with a representable matroid.

Network computation problems can be characterized using FD-relations.

Extension of the framework to nonlinear network codes.

Abstract

We consider the following \textit{network computation problem}. In an acyclic network, there are multiple source nodes, each generating multiple messages, and there are multiple sink nodes, each demanding a function of the source messages. The network coding problem corresponds to the case in which every demand function is equal to some source message, i.e., each sink demands some source message. Connections between network coding problems and matroids have been well studied. In this work, we establish a relation between network computation problems and representable matroids. We show that a network computation problem in which the sinks demand linear functions of source messages admits a scalar linear solution if and only if it is matroidal with respect to a representable matroid whose representation fulfills certain constraints dictated by the network computation problem. Next, we obtain a connection between network computation problems and functional dependency relations (FD-relations) and show that FD-relations can be used to characterize network computation problem with arbitrary (not necessarily linear) function demands as well as nonlinear network codes.