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, whereby the relays cooperate through degraded sets of message parts, achieves the cutset upper bound within $(1/2)\log N$ bits regardless of the channel gains and power constraints. This scheme improves upon a previous scheme by Chern and Ozgur, which is also based on partial decode--forward yet has an unbounded gap from the cutset bound for $L \ge 2$ destinations. When specialized to independent codes among relays, the proposed scheme achieves within $\log N$ bits from the cutset bound. The computation of this relaxation involves evaluating mutual information across $L(N+1)$ cuts out of the total $L 2^N$ possible cuts, providing a very simple linear-complexity algorithm to approximate the single-source multicast capacity of the Gaussian two-hop relay network.

7 pages (2 columns), 4 figures, submitted to the IEEE Transactions on Information Theory