A Note on the Multiple Unicast Capacity of Directed Acyclic Networks

We consider the multiple unicast problem under network coding over directed acyclic networks with unit capacity edges. There is a set of n source-terminal (s_i-t_i) pairs that wish to communicate at unit rate over this network. The connectivity between the s_i-t_i pairs is quantified by means of a connectivity level vector, [k₁ k₂ ... k_n] such that there exist k_i edge-disjoint paths between s_i and t_i. Our main aim is to characterize the feasibility of achieving this for different values of n and [k₁ ... k_n]. For 3 unicast connections (n = 3), we characterize several achievable and unachievable values of the connectivity 3-tuple. In addition, in this work, we have found certain network topologies, and capacity characterizations that are useful in understanding the case of general n.