  In a K-user Gaussian interference channel, it has been shown that if for each
user the desired signal strength is no less than the sum of the strengths of
the strongest interference from this user and the strongest interference to
this user (all values in dB scale), then treating interference as noise (TIN)
is optimal from the perspective of generalized degrees-of-freedom (GDoF) and
achieves the entire channel capacity region to within a constant gap. In this
work, we show that for such TIN-optimal interference channels, even if the
message set is expanded to include an independent message from each transmitter
to each receiver, operating the new channel as the original interference
channel and treating interference as noise is still optimal for the sum
capacity up to a constant gap. Furthermore, we extend the result to the
sum-GDoF optimality of TIN in the general setting of X channels with arbitrary
numbers of transmitters and receivers.
