A Note on the Multiple-Unicast Network Coding Conjecture

Network coding encourages in-network information mixing for enhanced network capacity. An important open problem in network coding, the multiple-unicast network coding (MUNC) conjecture, claims that network coding is equivalent to routing for multiple unicast sessions in an undirected network. Simple and intuitive as it appears, the MUNC conjecture has remained open for a decade. This work studies a special case of MUNC, where intra-session connections are "better" than inter-session connections. In the cost domain, we prove that if each source is closer to its own receiver than to other receivers, then network coding is equivalent to routing. In the throughput domain, an intuitive dual statement is that if each source has a larger max-flow to its own receiver than to other receivers, then network coding is equivalent to routing. We prove that this statement is equivalent to the original MUNC conjecture. An interesting consequence is that an equivalence between the two seemingly dual statements studied in this work, if true, would imply the original MUNC conjecture.