On the Multiple-Unicast Conjecture: Beyond Cut Metrics
TL;DR
A new proof that the multiple-unicast conjecture holds for networks with at most six coding nodes, without computer-aided search, is given, and it is shown that if the conjecture holds on $\Gamma_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.
Abstract
Network coding allows intermediate nodes to encode received messages before transmission. The multiple-unicast conjecture asserts that coding has no throughput advantage over fractional routing for independent unicast sessions in any undirected network. Despite more than two decades of sustained study, this central open problem remains unresolved. The conjecture is deeply connected to computational complexity: a proof would yield long-sought lower bounds for fundamental problems. To study the conjecture, this paper develops a unified metric framework from the perspective that the basic objects behind the comparison between coding and routing are not cuts alone, but graph metrics. Using this framework, we prove the conjecture for three new classes of undirected networks: (a) networks with at most five terminal locations; (b) planar networks whose terminal locations lie on the boundaries of at most three designated faces, with each session's endpoints on one such face; and (c) networks with arbitrarily many nodes and terminal locations under a structural restriction on session endpoints. We give a new proof that the conjecture holds for networks with at most six coding nodes, without computer-aided search, and show that if the conjecture holds on $\Gamma_{3,3}$, then it holds whenever no three sessions have six distinct terminal locations.