  The robust principles of treating interference as noise (TIN) when it is
sufficiently weak, and avoiding it when it is not, form the background of this
work. Combining TIN with the topological interference management (TIM)
framework that identifies optimal interference avoidance schemes, we formulate
a TIM-TIN problem for multilevel topological interference management, wherein
only a coarse knowledge of channel strengths and no knowledge of channel phases
is available to transmitters. To address the TIM-TIN problem, we first propose
an analytical baseline approach, which decomposes a network into TIN and TIM
components, allocates the signal power levels to each user in the TIN
component, allocates signal vector space dimensions to each user in the TIM
component, and guarantees that the product of the two is an achievable number
of signal dimensions available to each user in the original network. Next, a
distributed numerical algorithm called ZEST is developed. The convergence of
the algorithm is demonstrated, leading to the duality of the TIM-TIN problem
(in terms of GDoF). Numerical results are also provided to demonstrate the
superior sum-rate performance and fast convergence of ZEST.
