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.