Approximate capacity of Gaussian relay networks: Is a sublinear gap to the cutset bound plausible?

Beginning with work by Avestimehr, Diggavi and Tse, there have been a series of papers showing that the capacity of Gaussian relay networks can be closely approximated by the cutset bound. More precisely, it is known that the gap between the cutset bound and capacity in these networks can be bounded by a function that grows linearly with the number of nodes in the network and is otherwise independent of network topology and channel configurations. We argue that this linear gap is fundamental to such approximations, and prove that improvement to a sublinear function is possible if, and only if, capacity is equal to the cutset bound for all Gaussian relay networks.