  We consider the information-theoretic optimality of treating inter-cell
interference as noise (multi-cell TIN) in downlink cellular networks. We focus
on scenarios modeled by the Gaussian interfering broadcast channel (IBC),
comprising $K$ mutually interfering Gaussian broadcast channels (BCs), each
formed by a base station communicating independent messages to an arbitrary
number of users. We establish a new power allocation duality between the IBC
and its dual interfering multiple access channel (IMAC), which entails that the
corresponding generalized degrees-of-freedom regions achieved through
multi-cell TIN and power control (TINA regions) for both networks are
identical. As by-products of this duality, we obtain an explicit
characterization of the IBC TINA region from a previously established
characterization of the IMAC TINA region; and identify a multi-cell convex-TIN
regime in which the IBC TINA region is a polyhedron (hence convex) without the
need for time-sharing. We then identify a smaller multi-cell TIN regime in
which the IBC TINA region is optimal and multi-cell TIN achieves the entire
capacity region of the IBC, up to a constant gap. This is accomplished by
deriving a new genie-aided outer bound for the IBC, that reveals a novel
BC-type order that holds amongst users in each constituent BC (or cell) under
inter-cell interference, which in turn is not implied by previously known
BC-type orders (i.e. degraded, less noisy and more capable orders). The
multi-cell TIN regime that we identify for the IBC coincides with a
corresponding multi-cell TIN regime previously identified for the IMAC, hence
establishing a comprehensive uplink-downlink duality of multi-cell TIN in the
GDoF (and approximate capacity) sense.
