  It has been shown recently by Geng et al. that in a $K$ user Gaussian
interference network, 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 signal strengths measured in dB
scale), then power control and treating interference as noise (TIN) is
sufficient to achieve the entire generalized degrees of freedom (GDoF) region.
Motivated by the intuition that the deterministic model of Avestimehr et al.
(ADT deterministic model) is particularly suited for exploring the optimality
of TIN, the results of Geng et al. are first re-visited under the ADT
deterministic model, and are shown to directly translate between the Gaussian
and deterministic settings. Next, we focus on the extension of these results to
parallel interference networks, from a sum-capacity/sum-GDoF perspective. To
this end, we interpret the explicit characterization of the
sum-capacity/sum-GDoF of a TIN optimal network (without parallel channels) as a
minimum weighted matching problem in combinatorial optimization, and obtain a
simple characterization in terms of a partition of the interference network
into vertex-disjoint cycles. Aided by insights from the cyclic partition, the
sum-capacity optimality of TIN for $K$ user parallel interference networks is
characterized for the ADT deterministic model, leading ultimately to
corresponding GDoF results for the Gaussian setting. In both cases, subject to
a mild invertibility condition the optimality of TIN is shown to extend to
parallel networks in a separable fashion.
