A Graph Minor Perspective to Multicast Network Coding

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 paper 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 for multicasting two information flows is almost equivalent to the Hadwiger conjecture, which connects graph minors with graph coloring. Such equivalence implies the existence of K4, K5, K6, and KO(q/log q) minors, for networks that require F3, F4, F5, and Fq to multicast two flows, respectively. We finally prove that, for the general case of multicasting arbitrary number of flows, network coding can make a difference from routing only if the network contains a K4 minor, and this minor containment result is tight. Practical implications of the above results are discussed.