this bound is lower than that of the bound $\overline{C}_{\Sigma}$ obtained with the cut-set approach, which makes it tighter as $P$ increases.

Next, we provide achievability schemes for the Y-channel where we use complete decode-and-forward, and functional decode-and-forward.

Lower Bound: Complete Decode and Forward {#LowerBound1}
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We describe a complete decode and forward scheme for the Y-channel. In this scheme, user $j$ encodes his messages $m_{jk}$ and $m_{jl}$ into an i.i.d. sequence $x_j^n(m_{jk},m_{jl})$ where $X_j\sim\mathcal{N}(0,P)$. Then, all users transmit their signals to the relay together. The relay decodes all messages in a MAC fashion, with a small probability of error if the rate tuple ${\ensuremath{\mathbf{R}}}$ is in the capacity region from the 3 users to the relay. Hence, we get the following sum-rate constraint in the uplink $$\begin{aligned}
\label{MACconstraint}
R_\Sigma\leq C(h_1^2P+h_2^2P+h_3^2P).\end{aligned}$$

The relay decodes ${\ensuremath{\mathbf{m}}}$ from its received signal. Then it uses a Gaussian codebook to encode ${\ensuremath{\mathbf{m}}}$ into an i.i.d. sequence $x_r^n({\ensuremath{\mathbf{m}}})$ where $X_r\sim\mathcal{N}(0,P_r)$. The relay then sends $x_r^n({\ensuremath{\mathbf{m}}})$. After receiving a noisy observation of $x_r^n({\ensuremath{\mathbf{m}}})$, user 1 knowing $m_{12}$ and $m_{13}$ can decode all other messages as long as [@GunduzYenerGoldsmithPoor] $$\begin{aligned}
\label{DFC1}
R_{21}+R_{23}+R_{31}+R_{32}\leq C(h_1^2P_r).\end{aligned}$$ Similarly at the other receivers, reliable decoding is guaranteed if the following rate constraints 