## 2 Related Works  
The purpose of this paper is to solve the joint relay selection, channel assignment and power optimization problem by using the pricing-based matching algorithm. In [31; 32; 33; 34], the proper relay stations were selected to improve the performance of the multi-hop D2D network while simultaneously considering the co-channel interference. In [35], the authors presented a theoretical framework for multi-hop D2D communications, which enables modeling of interference and performance analysis. In [36], the authors proposed a novel routing protocol for underwater sensor networks (UWSNs), which was named the location-aware routing protocol (LARP) and could improve packet delivery ratio and reduce normalized routing overhead. In [37], the authors modeled the problems of relay selection and channel allocation as a classical weighted bipartite graph matching problem and solved it by using the Hungarian algorithm. In [38], a novel quantum particle swarm optimization-based relay selection scheme was proposed to maximize the system throughput of the cooperative relay networks and reduce the computational complexity compared with the exhaustive search. The above works mainly focus on how to maximize the throughput of the system through relay selection with the presence of interference. In particularly, the channel allocation problem was also considered in [37]. In [39; 40; 41; 42; 43], the resource allocation problem was solved simultaneously with the relay selection problem. In [20], the joint resource allocation and relay selection problem was formulated as a binary integer non-linear programming problem, and the optimal solution of the random linear network coding aided D2D communications was obtained by introducing the concept of D2D cluster. In [44], an algorithm was proposed to maximize the throughput of a D2D pair by finding the optimum multi-hop routes and the corresponding transmit powers. To achieve simultaneous downlink transmission for multiple users, in [45], the authors proposed a novel MAC scheme based on an opportunistic channel-aware scheduling policy. In [46], the authors employed the Lyapunov optimization theory to decompose the dynamic communication mode selection problem and provide a lower bound of the system throughput and proposed an algorithm to achieve approximate maximization. In [47], the joint relay selection and resource allocation problem were solved in a multi-user, multi-carrier and multi-cellular network, thus increasing the throughput of users in the cell-edge. As the co-channel interference exists in the underlying mode of the D2D communication network, the resource allocation problem must be solved under the interference constrains to guarantee the quality of communications [48; 49]. In [12], a three-dimensional matching algorithm was applied to maximize the weighted channel rate of the system, which explored the information from both the social layer and the physical layer. However, the EE performance was neglected. To extend the lifetime of the rapidly-drained battery of user equipment, when matching D2D users with relays, it is necessary to consider the EE [26; 27; 50]. In [51], the selection of the energy harvesting relay has been studied for D2D communications in a public safety environment. In [23], the iterative Hungarian method was devised to solve the associated relay selection and resource assignment problems, and it showed that the optimal power allocation problem can be solved in a closed-form. In [52], the authors have proven that threshold-based power control policies function efficiently, and this would benefit the limited battery of users' handheld equipment. However, most of the above works have not considered EE of D2D links during the optimization process. Although in [26; 51], the EE issue has been considered, the multi-hop D2D communication was neglected, and the application scenario is different from our study.