## 3 System Model and Problem Formulation
### Problem Formulation
#### 3.2.3 Throughput Optimization  
Since, we presumed that the eNB has global CSI of all the available CUs and DPs. Thus, the system can handle a peak power level. Therefore, the sum throughput optimization problem of the network can be formulated as follows:  
\[\text{A1.arg}\max_{m\;cD,\;k\in K}\sum_{n=1}^{N}F\left(log_{2}\left(1+\frac{P_{n }^{k}l_{n,B}^{-\alpha}|h_{n,B}|^{2}}{\sigma_{n}^{2}}\right)+\sum_{n=1}^{C}G_{n}\right) \tag{26}\]  
subject to:  
\[\frac{P_{n}^{k}l_{n,B}^{-\alpha}|h_{n,B}|^{2}}{I_{n}^{m_{t}}+I_{n}^{m_{t}}+{ \sigma_{n}}^{2}}\geq\gamma_{n_{min}}^{k},\;\forall n\in C,\;m_{t},m_{t}^{\prime }\in D,k\in X, \tag{27}\]  
\[\frac{\delta_{m,n}^{k}\cdot P_{m_{t}}^{k}\cdot l_{m_{t},m_{t}}^{-\alpha_{t}}|h _{m_{t},m_{t}}|^{2}}{I_{m_{t}}^{n}+I_{m_{t}}^{n}+I_{m_{t}}^{m_{t}}+{\sigma_{n }}^{2}}\geq\gamma_{n_{min}}^{k},\;\forall n\in C,\;m_{t},m_{t}^{\prime}\in D,k \in X, \tag{28}\]  
\[P_{n}^{min}\;\leq\;P_{n}^{k}\leq P_{n}^{max},\;\forall n\in C,k\in X, \tag{29}\]  
\[P_{m_{t}}^{min}\;\leq\;P_{m_{t}}^{k}\leq P_{m_{t}}^{max},\;\forall m_{t}\; \epsilon\;D,\;k\in X, \tag{30}\]  
\[\delta_{m,n}^{k}\;\epsilon\;\{0,1\},\;\forall\;m\;\epsilon\;D,\;n\;\epsilon\; C,\;k\in X, \tag{31}\]  
where \(\gamma_{n_{Th}}^{k}\) and \(\gamma_{n_{Th}}^{k}\) denote the minimum SINR requirements of CU \(n\) and DP \(m\) using channel \(k\), respectively. \(P_{n}^{min}\) and \(P_{m_{t}}^{min}\) are the minimal transmit power of CU \(n\) and D2D transmitter \(m_{t}\), respectively. \(P_{n}^{max}\) and \(P_{m_{t}}^{max}\) represent the maximal transmit power of CU \(n\) and \(P_{m_{t}}^{max}\), respectively. The constraints in Equations (27) and (28) imply that the SINRs of both CUs and DPs should be equal to or more than the minimum required SINR to achieve the QoS requirements. The constraints in Equations (29) and (30) guarantee that the transmit power of CUs and DPs should be within the upper and lower bound power levels. The constraint in Equation (31) guaranteed the versatility of resource reuse factor within the upper and lower bounds power level. This assures QoS requirements of the users by limiting the uplink interference.  
However, during the CSI exchange between large number of users and eNB, a significant interference overhead is generated. From the optimization problem formulated in Equation (26), we can see that the problem statement aimed to find a global optimal solution with an exhaustive search. Therefore, we analyze the computational complexity of the throughput maximization problem. First, we calculate the throughput of traditional CUs before the integration of D2D communications. Therefore, the throughput maximization problem has computational complexity of \(O(N\times K)\). Finally, we allocate the subcarrier resources to DPs, such that the resource reuse factor is greater than one. Hence, the complexity of the final step is \(O(M\times K\times F)\). Therefore, the overall computational complexity of the network is \(O(N\times K)+O(M\times K\times F)\). We observed that the computational complexity of the network is very high, which is not practical. Moreover, the computational complexity of a network increases exponentially with problem size. Thus, a low-complexity approach suitable for practical D2D communications environment to reduce the interference overhead is proposed in Section 4.