  Consider a wireless Gaussian network where a source wishes to communicate
with a destination with the help of N full-duplex relay nodes. Most practical
systems today route information from the source to the destination using the
best path that connects them. In this paper, we 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. More
specifically, we show that for $N$-relay Gaussian networks with an arbitrary
topology, routing can in the worst case guarantee an approximate fraction
$\frac{1}{\left\lfloor N/2 \right\rfloor + 1}$ of the capacity of the full
network, independently of the SNR regime. We prove that this guarantee is
fundamental, i.e., it is the highest worst-case guarantee that we can provide
for routing in relay networks. Next, we consider how these guarantees are
refined for Gaussian layered relay networks with $L$ layers and $N_L$ relays
per layer. We prove that for arbitrary $L$ and $N_L$, there always exists a
route in the network that approximately achieves at least $\frac{2}{(L-1)N_L +
4}$ $\left(\mbox{resp.}\frac{2}{LN_L+2}\right)$ of the network capacity for odd
$L$ (resp. even $L$), and there exist networks where the best routes exactly
achieve these fractions. These results are formulated within the network
simplification framework, that asks what fraction of the capacity we can
achieve by using a subnetwork (in our case, a single path). A fundamental step
in our proof is a simplification result for MIMO antenna selection that may
also be of independent interest. To the best of our knowledge, this is the
first result that characterizes, for general wireless network topologies, what
is the performance of routing with respect to physical layer cooperation
techniques that approximately achieve the network capacity.
