On an equivalence of the reduction of k-unicast to 2-unicast capacity and the edge removal property

Abstract

In recent work, Kamath et al. show that network code design for any k-unicast network reduces to network code design for a related 2-unicast network. The proof assumes that codes achieve their desired rates precisely (rather than approaching them asymptotically) and that error probability equals zero. We study two questions posed in but left unanswered by the Kamath et al. paper. The first asks whether the reduction for 0-error code design can be extended to show an equivalence in 0-error network capacity, which includes rates approached asymptotically. The second asks whether the reduction can be generalized to show an equivalence in Shannon capacity, which requires that error probability approach (but not necessarily hit) zero. While we do not solve these questions, we show that finding the k-unicast capacity reduces to finding the 2-unicast capacity under this reduction if and only if the so called 'edge removal statement' is true for all networks. This equivalence holds under both 0-error and asymptotic notions of reliability.