Does network coding improve the throughput of a survivable multicast network?

We address survivability considerations for telecommunication networks where a part of the network may fail. We focus on single arc failures in multicast networks, with or without network coding. The problem is to compute a routing such that, if any single arc failure occurs, the remaining throughput is as large as possible. In the case of multicast routing without network coding, two ways for computing the impact of an arc failure are considered. Either the throughput of entire Steiner trees containing that arc vanishes, or only the throughput of subtrees, starting from that arc to terminals, is discarded. In the case of multicast with network coding, since the total throughput is computed as the minimum of single max-flow values from the source to any terminal, we consider that a failure of an arc implies that the throughput of all paths containing that arc vanishes. The three obtained models are formulated as linear problems. We solve these problems either directly or by use of column generation. Further, the survivable network coding gain is defined as the ratio of maximal throughput with - over without - network coding, when arc failures can occur. We show that this ratio is unbounded for unitary combination digraphs and hence for any directed networks. We also show that this ratio is equal to one for bidirected graphs with uniform capacities. Finally, some numerical results are carried out on randomly generated networks showing that the gain ratio is equal to one for almost 98 percent of the instances. However, we observe that routing with network coding requires much less redundancy in order to achieve the optimal throughput.