Wireless Network Simplification: The Performance of Routing

This paper explores the <italic>network simplification</italic> problem for Gaussian full-duplex relay networks with arbitrary topology. Particularly, given an <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-relay Gaussian full-duplex network, the network simplification problem seeks to find fundamental guarantees on the capacity of the best subnetwork, among a particular class of subnetworks, as a fraction of the full-network capacity. The focus of this work is the case when the selected subnetwork class is a path from the source to the destination. The main result of the paper shows that for an <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-relay Gaussian networks with arbitrary topology, the best route can in the worst case guarantee an approximate fraction <inline-formula> <tex-math notation="LaTeX">$\frac {1}{ \lfloor N/2 \rfloor + 1}$ </tex-math></inline-formula> of the capacity of the full network, independently of the channel coefficients and/or operating SNR. Furthermore, this guarantee is shown to be fundamental, i.e., it is the highest worst-case guarantee that can be provided for routing in relay networks. A key step in the proof of the main result lies in the derivation of a simplification result for antenna selection in MIMO channels that may also be of independent interest. To the best of our knowledge, this is the first result that characterizes the performance of routing in comparison to physical layer cooperation techniques that approximately achieve the network capacity for general wireless network topologies. The results in this paper show that routing can, in the worst case, result in an unbounded gap from the network capacity - or reversely, physical layer cooperation can offer unbounded gains over routing.