Network coding in minimal multicast networks

We investigate the network coding problem in a certain class of minimal multicast networks. In a multicast coding network, a source S needs to deliver h symbols, or packets, to a set of destinations T over an underlying communication network modeled by a graph G. A coding network is said to be h-minimal if it can deliver h symbols from S to the destination nodes, while any proper subnetwork of G can deliver at most h - 1 symbols to the set of destination nodes. This problem is motivated by the requirement to minimize the amount of network resources allocated for a multicast connections. We show that, surprisingly, minimal multicast networks have unique properties that distinguish them from the general case of multicast networks. In particular, we show that it is possible to determine whether a 2-minimal network has a routing solution (i.e., a solution without encoding nodes) in polynomial time, while this problem is NP-hard in general. In addition, we show that if a 2-minimal network is planar, then the minimum size of the required field for linear network codes is at most 3. Also, we investigate several structural properties of 2-minimal networks and generalize our results for h > 2.