The multicast capacity of the Gaussian two-hop relay network with one source, N relays, and L destinations is studied. It is shown that a careful modification of the partial decode-forward coding scheme, in which the relays cooperate through degraded sets of message parts, achieves the cutset upper bound within log N bits regardless of the channel gains and power constraints. This scheme improves upon a previous scheme by Chern andÖzgür, which is also based on partial decodeforward yet has an

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... bounded gap from the cutset bound for L ≥ 2 destinations. As an added benefit, the achievable rate of the proposed scheme involves evaluating rate for L(N + 1) cuts out of the total L2 N possible cuts.