In D'Angelo et al. [28] and in Audrito et al. [29], the authors study the bandwidth constraint problem. This concerns flow on multi-interface networks. In these papers, each interface is associated with a specific bandwidth. The Maximum Flow and Minimum Cost Flow problems aim at finding a connection between two nodes by considering the bandwidth constraints. The Maximum Flow problem aims at maximising the bandwidth between the two nodes. All the network interfaces are assumed to be active. In this way all the permitted connections can be established. In this context, a flow function that guarantees the maximum communication bandwidth between the two nodes has been studied. This problem is a generalisation of the Maximum Flow problem for standard networks. The Minimum Cost Flow problem aims at finding a communication sub-network between two nodes that has a minimum energy consumption cost and ensures at least a given communication bandwidth B. More precisely, the problem finds the set of active interfaces that has minimum cost and ensures that a certain node s can exchange **data** with another node t with a bandwidth not smaller than B. Although the solution can be a path between s and t, a more complex graph can be required. This has nodes with active interfaces that depend on the network topology. This problem is a generalisation of the Minimum Cost Flow problem for standard networks.