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In this paper we study the Gaussian relay-interference network, in which relay (helper) nodes are to facilitate competing information flows over a wireless network. We examine this problem for certain regimes of channel values, when one of the cross-links is dominated by noise, resulting in Z and/or S configurations for the networks. For these Gaussian ZZ and ZS networks, we establish an approximate characterization of the rate region. The outer bounds to the capacity regions are established using genie-aided techniques that extend the methods used for the Gaussian interference channel to the relay-interference network. For the inner bound of the ZZ network, we utilize a new interference management scheme, termed interference neutralization, which was inspired by our earlier study of such deterministic networks. This technique allows for over-the-air interference removal, without the transmitters having complete access to the interfering signals.