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

Partial Decode-Forward Relaying for the Gaussian Two-Hop Relay Network