A Graph Minor Perspective to Multicast Network Coding

TL;DR

This paper explores the deep connection between network coding requirements and the underlying graph topology, establishing that certain network coding benefits are linked to specific graph minors, with implications for network design and analysis.

Contribution

It introduces the NC-Minor Conjecture linking graph minors to network coding necessity and proves its near equivalence to the Hadwiger Conjecture, revealing topological conditions for coding benefits.

Findings

Network coding benefits are tied to the presence of specific graph minors.

Networks requiring certain finite fields must contain corresponding minors.

Network coding can outperform routing only if the network contains a $K_4$ minor.

Abstract

Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often treats the network topology as a black box, focusing on algebraic or information theoretic aspects of the problem. This work aims at an in-depth examination of the relation between algebraic coding and network topologies. We mathematically establish a series of results along the direction of: if network coding is necessary/beneficial, or if a particular finite field is required for coding, then the network must have a corresponding hidden structure embedded in its underlying topology, and such embedding is computationally efficient to verify. Specifically, we first formulate a meta-conjecture, the NC-Minor Conjecture, that articulates such a connection between graph theory and network coding, in the language of graph minors. We next prove that the NC-Minor Conjecture is almost equivalent to the Hadwiger Conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of $K_4$, $K_5$, $K_6$, and $K_{O(q/\log{q})}$ minors, for networks requiring $\mathbb{F}_3$, $\mathbb{F}_4$, $\mathbb{F}_5$ and $\mathbb{F}_q$, respectively. We finally prove that network coding can make a difference from routing only if the network contains a $K_4$ minor, and this minor containment result is tight. Practical implications of the above results are discussed.