This paper considers a Gaussian relay-interference channel and introduces a generalized hash-and-forward relay strategy, where the relay sends out a bin index of its quantized observation, and the receivers first decode the relay quantization codeword to a list, then use the list to help decode the respective messages from the transmitters. The main advantage of the proposed approach is in a scenario where the relay observes a linear combination of the transmitted signals and broadcasts a common relay message through a digital relay link of fixed rate to help both receivers of the interference channel. We show that when compared to the achievable rates with interference treated as noise, generalized hash-and-forward can provide one bit of rate improvement for every relay bit for both users at the same time in an asymptotic regime where the background noises go down to zero. The proposed approach is universal, in contrast to the compress-and-forward or amplify-and-forward strategies which are not asymptotically optimal for multiple users simultaneously, if at all.