## 2 System Model  
We consider a single-cell multicast system where an eNB serves numerous UEs and the UEs in close proximity can group into a cluster to request the same data from eNB. The UEs within a cluster can not only communicate with eNB, but also directly communicate with each other via D2D links. Let us focus on one D2D cluster, as illustrated in Figure 1. Due to independent properties of cellular downlink channels, when the eNB multicasts data to all UEs within a D2D cluster at a certain rate, only a portion of the UEs can correctly receive and decode the data. The UEs that can and cannot successfully receive data are referred to as \"ACK-UEs\" and \"NACK-UEs\", respectively. In order to share the data from eNB in the whole cluster, ACK-UEs can be employed as relays to forward the received data to NACK-UEs via D2D links. In general, the underlined intra-cluster relay transmissions can be accomplished via D2D multicast. Meanwhile, to avoid the interference between D2D communication and cellular communication, orthogonal resources should be allocated to them. In the paper, we only consider both large-scale path-loss attenuation and smallscale fading characterized by Rayleigh fading, and thus the channel response follows the independent complex  
Figure 1: D2D clusters with ACK-UEs and NACK-UEs in the cellular network. Gaussian distribution. In addition, the free space propagation path-loss model, \(P=P_{\text{\tiny${i}$}}\cdot\left(d\,/\,d_{\text{\tiny${0}$}}\right)^{-\alpha}\), is used where \(P_{\text{\tiny${0}$}}\) and \(P\) represent signal power measured at \(d_{\text{\tiny${0}$}}\) and \(d\) meters away from the transmitter, respectively. \(\alpha\) is a path-loss exponent. We simplify the received power at \(d_{\text{\tiny${0}$}}=1\) equals the transmit power. Hence, the received power of each D2D link can be expressed as. \[P_{\text{\tiny${i}$},\text{\tiny${j}$}}=P_{\text{\tiny${i}$}}\cdot h_{\text{ \tiny${j}$}}^{2}=P_{\text{\tiny${i}$}}\cdot\left(d_{\text{\tiny${j}$}}\right)^{ -\alpha}\cdot h_{\text{\tiny${0}$}}^{2} \tag{1}\]  
where \(P_{\text{\tiny${i}$},\text{\tiny${j}$}}\) and \(d_{\text{\tinyFigure 1:0}}\) are the received power and the distance of the \(i\)-\(j\) link. \(P_{\text{\tinyFigure 1:1}}\) represents the transmit power of UE \(i\), and \(h_{\text{\tinyFigure 1:2}}\) is the complex Gaussian channel coefficient that obeys the distribution \(\mathcal{CN}\left(0,1\right).\) Given the power spectral density of the additive Gaussian noise \(\sigma^{\text{\tinyFigure 1:3}}\) and system bandwidth \(W_{\text{\tinyFigure 1:4}}\), according to Shannon formula, the channel rate of UE \(j\) can be obtained by. Figure 1:5  
We define \(\mathcal{N}_{\text{\tiny{NACK}}}\) is the set of NACK-UEs and \(\mathcal{N}_{\text{\tiny{ACK}}}\) is the set of ACK-UEs. The numbers of NACK-UEs and ACK-UEs in the cluster are denoted as \(N_{\text{\tiny{NACK}}}\) and \(N_{\text{\tiny{ACK}}}\), respectively. Besides, inspired by [7], we assume there are \(L\) ACK-UEs acting as relays. For a given relay, the achievable D2D multicast rate depends on the worst link among all links connecting it with all NACK-UEs. By optimizing selecting of a D2D relay, the system sum data rate can be defined as  
\[\mathcal{R}_{\text{\tiny{opt}}}=\max_{ \begin{subarray}{c}n_{i}\in\mathcal{N} _{\text{\tiny{ACK}}},n_{1}\
eq n_{2}\
eq\cdots n_{L}\end{subarray} }\sum_{i=1} ^{L}\sum_{j\
eq\text{\tiny{ACK}}}\min_{ \begin{subarray}{c}n_{i}\in\mathcal{N} _{\text{\tiny{ACK}}},n_{\text{\tiny{ACK}}}\end{subarray} }R_{\text{\tiny{ \tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{\tiny{ \tinytiny{\tinytiny{\tinytiny{\tinytinytinytiny{\tinytiny{     \tiny{\tiny{{\tiny            {{           }}}}}}}}}}}}}  
which an element package is expressed as \(\mathcal{D}_{n}\,,m=1,2,...,M\) in correspondence with the \(m\)-th ACK-UE. The possible packages are subsets of NACK-UEs. By using \(\mathcal{D}\), we can transform the optimization problem in Equation (3) into a package assignment process.