## 1 Introduction  
### Background and Motivation  
As the demand for wireless data services grows dramatically, revolutionary technologies that focus on improving network coverage and throughput within the limited system resources have been constantly emerging. Device-to-device (D2D) communication that enables user equipment (UEs) to communicate with each other through direct links without the base station (BS) relaying [1] is considered to be one of the key technologies in the future 5G wireless system [2]. In the underlay mode, D2D links share the same spectrum resources with traditional cellular UEs (CUEs), providing significant increases in spectrum efficiency and system throughput [1; 3]. There are two basic transmission types in D2D communication: single-hop transmission, in which data are transmitted from the transmitter to the receiver directly as the channel condition between them meets the quality of service (QoS) requirement; and multi-hop transmission, where some devices act as relay stations, in which data are transmitted from the transmitter to the receiver hop by hop, and the route must be opportunistically optimized to guarantee the performance [4]. In general, because of the high channel quality of short-range D2D links, the D2D UEs can achieve a high single-hop transmission rate [5; 6]. However, there exists a possibility that the channel condition between two potential D2D UEs cannot support a direct single-hop link. For this reason, D2D cooperative relay has naturally become an important application that can assist two UEs with an inferior direct link to communicate [7; 8; 9]. Though applying underlying mode D2D communication in cellular networks can bring some advancements such as spectrum efficiency and the coverage of communication, the UEs can no longer neglect the co-channel interference caused by spectrum resource reusing [10; 11]. Since the underlay mode D2D communication brings new problems and challenges to the cellular networks, one of the crucial issues is the intelligent resource allocation strategy for both D2D UEs and CUEs to guarantee their QoS [12; 13; 14]. Due to the low cost and the few modifications to current infrastructures, the D2D relay may work more efficiently than fixed relays [15]. Nonetheless, the co-channel interference caused by spectrum resource sharing exists in D2D cooperative relay transmissions, which is identical to the single-hop transmission. As the relay communication requires two-hop spectrum resources, the resource management has become more complicated than in the single-hop case. Moreover, relay selection needs to be optimized jointly with the spectrum allocation and power control for both the first hop and the second hop links, which can realize the system performance maximization. There have already been some works that focus on relay selection [16; 17] and resource allocation [18; 19; 20; 21; 22] to optimize the performance of D2D relay communications. In [20], the joint resource allocation and relay selection problem was formulated as a binary integer nonlinear programming problem, and the authors in [21] also considered the joint optimization and proposed a two-stage method to maximize system throughput. However, few works above have covered the power control problem, which is closely related to energy efficiency (EE). As the booming wireless services consume a large amount of energy from UE batteries and UEs are handheld equipment with limited battery capacity, one key issue is to reduce the energy consumption for both D2D transmitters (TXs) and D2D relay stations (RSs). An iterative Hungarian method was proposed to solve the relay and spectrum allocation problem with optimal transmit power at each node in [23]. Nevertheless, EE has not been considered in this work, and the cellular spectrum resources are allowed to be reused by only one of the two hops in the D2D relay link. Therefore, we put emphasis on energy-efficient resource management in D2D cooperative relay communications.