Reduction and Fixed Points of Boolean Networks and Linear Network Coding Solvability

TL;DR

This paper explores the fixed points of Boolean networks and their application to linear network coding solvability, providing new classifications and bounds for solvable network configurations.

Contribution

It introduces a variable reduction method for coding functions and applies it to classify and identify solvable and non-solvable network graphs.

Findings

Non-decreasing coding functions do not outperform routing.

Triangle-free undirected graphs are linearly solvable iff solvable by routing.

Identifies new non-linearly solvable graphs and classes of linearly solvable graphs.

Abstract

Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this problem to determining the number of fixed points of coding functions $f:A^n\to A^n$ over a finite alphabet $A$ (usually referred to as Boolean networks if $A = \{0,1\}$) with a given interaction graph, that describes which local functions depend on which variables. In this paper, we generalise the so-called reduction of coding functions in order to eliminate variables. We then determine the maximum number of fixed points of a fully reduced coding function, whose interaction graph has a loop on every vertex. Since the reduction preserves the number of fixed points, we then apply these ideas and results to obtain four main results on the linear network coding solvability problem. First, we prove that non-decreasing coding functions cannot solve any more instances than routing already does. Second, we show that triangle-free undirected graphs are linearly solvable if and only if they are solvable by routing. This is the first classification result for the linear network coding solvability problem. Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we determine large classes of strictly linearly solvable graphs.