We consider wireless relay networks where a source node communicates to a

destination node with the help of multiple intermediate relay nodes. In

wireless, if a node can send information to another node, typically it can also

receive information from that node. Therefore, there are inherently a lot of

opportunities for feeding back information in such networks. However,

transmissions are not isolated and usually subject to broadcast and

interference.

In this paper, we ask the following question: Can the information transfer in

both directions of a link be critical to maximizing the end-to-end

communication rate in such networks? Equivalently, could one of the directions

in each bidirected link (and more generally at least one of the links forming a

cycle) be shut down and the capacity of the network still be approximately

maintained? Our main result is to show that in any arbitrary Gaussian relay

network with bidirected edges and cycles, we can always identify a directed

acyclic subnetwork that approximately maintains the capacity of the original

network. The edges of this subnetwork can be identified as the information

carrying links, and the remaining links as feedback, which can only provide

limited contribution to capacity.