## 3 Problem Formulation
\[\sum_{k=1}^{K}\sum_{i=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{7}\]  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]  
\[x_{i}^{k}\leq z_{i,j}^{k},\ \forall i,j=1,2,\ldots,N \tag{9}\]  
\[\sum_{i=1}^{N}x_{i}^{k}=\sum_{j=1}^{N}x_{i,j}^{k}\forall k=1,2,\ldots,K\,\ \forall u=1,2,\ldots,N,\ u\
eq i,j \tag{10}\]\[D^{j,k,j}_{delay}\leq\ D^{k}_{th},\forall k=1,2,\ldots,K\,\ \forall i,j=1,2,\ldots,N \tag{11}\]  
\[x^{k}_{ij}\in\ \{0,1\},\forall k=1,2,\ldots,K\,\ \forall i,j=1,2,\ldots,N \tag{12}\]  
\[z^{k}_{ij}\in\ \{0,1\},\forall k=1,2,\ldots,K\,\ \forall i,j=1,2,\ldots,N \tag{13}\]  
\[0\leq\alpha,\beta<1 \tag{14}\]  
\[\alpha+\beta=1 \tag{15}\]  
where \(D^{k}_{\text{th}}\) is the delay constraint for content \(k\); \(z^{k}_{ij}\) and \(x^{k}_{ij}\) are indicator functions. \(z^{k}_{i,j}\) is 1 if the Signal to Interference plus Noise Ratio (SINR) of the path from UE \(i\) to UE \(j\) meets the minimal requirement of content \(k\), and it is 0 otherwise. \(x^{k}_{i,j}\) is 1 if UE \(i\) downloads content \(k\) from UE \(j\) with a satisfied SINR ratio (\(z^{k}_{i,j}=1\)), and it is 0 otherwise. Equations (7) and (8) indicate that content \(k\) will pass through UE \(i\) and UE \(j\) at least once. Equation (9) is to ensure that the SINR of the path from UE \(i\) to UE \(j\) meets the minimal requirement of content \(k\). Equation (10) implies the conservation of flow. UE \(u\) is an intermediate node on the path between UE \(i\) and UE \(j\). Any content entering UE \(u\) will be forwarded toward its destination. Equation (11) shows that the time UE \(i\) takes in downloading content \(k\) from UE \(j\) must meet the delay constraint of content \(k\). The routing problem for group communication defined in Equation (6) is an NP-hard problem. It is equivalent to the Open Vehicle Routing Problem (OVRP), which had been proved to be an NP-hard problem. OVRP originated from the Vehicle Routing Problem (VRP) introduced by [PERSON] and [PERSON] in 1959 [31]. The difference for OVRP is that vehicles need not return to the service center after delivery. In a VRP, there are several customers, each having merchandise needs. A distribution center serves customers with a vehicle fleet. The object is to plan the routes for vehicles with minimal total cost. An OVRP can be formulated as follows:  
\[min\sum_{k=1}^{K}\sum_{i=1}^{N}\sum_{j=1}^{N}C^{k}_{i,j}x^{k}_{i,j} \tag{16}\]  
s.t. \[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]0  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]1  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]2  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]3  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]4  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]5  
\[\sum_{k=1}^{K}\sum_{j=1}^{N}x_{i}^{k}\geq 1,\ \forall i=1,2,\ldots,N,\ i\ \
eq\ j \tag{8}\]6  
where \(i\) and \(j\) are indices for customers; \(k\) is the index for vehicles; \(C^{k}_{i,j}\) is the cost it takes for vehicle \(k\) to travel from customer \(i\) to customer \(j\). \(x^{k}_{i,j}\) and \(z^{k}_{ij}\) are indicator functions. \(x^{k}_{i,j}\) is 1 if vehicle \(k\) takes the path from customer \(i\) to customer \(j\), and it is 0 otherwise. \(z^{k}_{i,j}\) is 1 if vehicle \(k\) can pass through customer \(i\) and customer \(j\), and it is 0 otherwise. \(G_{k}\) is the capacity of vehicle \(k\). \(q^{k}_{i}\) is the weight of customer \(i\)'s merchandise on vehicle \(k\). Equations (17) and (18) indicate that vehicle \(k\) will pass through customer \(i\) and customer \(j\) at least once. Equation (19) is to ensure that the vehicle serving customer \(i\) and customer \(j\) must be valid. Equation (20) implies the conservation of flow.