Path Gain Algebraic Formulation for the Scalar Linear Network Coding Problem

TL;DR

This paper introduces a novel algebraic method for solving scalar linear network coding problems using path gains, simplifying the polynomial equations to degree 2 and enabling more efficient solvability analysis.

Contribution

It proposes a path gain formulation that reduces polynomial equations to degree 2, facilitating simpler analysis and solution of network coding problems.

Findings

Path gain approach yields simpler polynomial equations.

The method determines solvability in small networks.

Successfully applied to a large network with 87 nodes.

Abstract

In the algebraic view, the solution to a network coding problem is seen as a variety specified by a system of polynomial equations typically derived by using edge-to-edge gains as variables. The output from each sink is equated to its demand to obtain polynomial equations. In this work, we propose a method to derive the polynomial equations using source-to-sink path gains as the variables. In the path gain formulation, we show that linear and quadratic equations suffice; therefore, network coding becomes equivalent to a system of polynomial equations of maximum degree 2. We present algorithms for generating the equations in the path gains and for converting path gain solutions to edge-to-edge gain solutions. Because of the low degree, simplification is readily possible for the system of equations obtained using path gains. Using small-sized network coding problems, we show that the path gain approach results in simpler equations and determines solvability of the problem in certain cases. On a larger network (with 87 nodes and 161 edges), we show how the path gain approach continues to provide deterministic solutions to some network coding problems.