  Aligned-Images Sumset Inequalities are used in this work to study the
Generalized Degrees of Freedom (GDoF) of the symmetric layered multi-hop
interference channel under the robust assumption that the channel state
information at the transmitters (CSIT) is limited to finite precision. First,
the sum-GDoF value is characterized for the $2\times 2\times 2$ setting that is
comprised of $2$ sources, $2$ relays, and $2$ destinations. It is shown that
the sum-GDoF do not improve even if perfect CSIT is allowed in the first hop,
as long as the CSIT in the second hop is limited to finite precision. The sum
GDoF characterization is then generalized to the $2\times 2\times \cdots \times
2$ setting that is comprised of $L$ hops. Remarkably, for large $L$, the GDoF
value approaches that of the one hop broadcast channel that is obtained by full
cooperation among the two transmitters of the last hop, with finite precision
CSIT. Previous studies of multi-hop interference networks either identified
sophisticated GDoF optimal schemes under perfect CSIT, such as aligned
interference neutralization and network diagonalization, that are powerful in
theory but too fragile to be practical, or studied robust achievable schemes
like classical amplify/decode/compress-and-forward without claims of
information-theoretic optimality. In contrast, under finite precision CSIT, we
show that the benefits of fragile schemes are lost, while a combination of
classical random coding schemes that are simpler and much more robust, namely a
rate-splitting between decode-and-forward and amplify-and-forward, is shown to
be GDoF optimal. As such, this work represents another step towards bridging
the gap between theory (optimality) and practice (robustness) with the aid of
Aligned-Images Sumset Inequalities.
