A reduction approach to the multiple-unicast conjecture in network coding

The multiple-unicast conjecture in network coding states that for multiple unicast sessions in an undirected network, network coding has no advantage over routing in improving the throughput or saving bandwidth. In this work, we propose a reduction method to study the multiple-unicast conjecture, and prove the conjecture for a new class of networks that are characterized by relations between cut-sets and source-receiver paths. This class subsumes the two known types of networks with non-zero max-flow min-cut gaps. Further combing this result with a computer-aided search, we derive as a corollary that network coding is unnecessary in networks with up to 6 nodes. We also prove the multiple-unicast conjecture for almost all unit-link-length networks with up to 3 sessions and 7 nodes.