On the advantage of network coding for improving network throughput

Given a data network with link capacities, we consider the throughput of the network for a multicast session involving a source node and a given set of terminals. It is known that network coding can improve the throughput of the network. We study the coding advantage, i.e. the ratio of the throughput using network coding to that without using network coding. We show that the maximum coding advantage for a given network is equal to the integrality gap of certain linear programming (LP) formulations for a Steiner tree. This holds for both directed as well as undirected networks. For directed networks, the coding advantage is equal to the integrality gap of the directed Steiner tree LP formulation; for undirected networks, the coding advantage is equal to the integrality gap of the bidirected cut LP formulation for the Steiner tree. This relates the coding advantage to well studied notions in combinatorial optimization. Further, this connection improves the known bounds on the coding advantage for both undirected as well directed networks.