On the separability of GDoF region for parallel Gaussian TIN optimal interference networks

It has been shown recently by Sun et al. that in a K user parallel Gaussian interference network, if over each sub-channel, 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 separate coding over each sub-channel and treating interference as noise (TIN) is sufficient to achieve the sum generalized degrees of freedom (GDoF), subject to a mild invertibility condition [1]. In this work, we show that the weighted sum GDoF is similarly separable, i.e., separate coding and TIN is sufficient to achieve the weighted sum GDoF, subject to a similar mild invertibility condition. This is proved by translating the weighted GDoF optimization problem to the sum GDoF problem of a class of compound parallel Gaussian interference networks, giving rise to new weighted GDoF outer bounds that are strictly stronger than what is implied by the sum GDoF bounds obtained previously.