## 1 Introduction  
The fifth generation (5G) cellular networks suffering from severe propagation issues massively promote the device-to-device (D2D) communication paradigm, i.e., direct connections between users, to improve the communications reliability at the network edges [1; 2; 3]. In the cooperative communications, already included in the previous fourth generation Long Term Evolution-Advanced cellular networks, fixed low-power nodes are arranged to support communication between the base station and the end-devices performing spatial diversity gains. As widely proved in the literature [1; 4; 5], the introduction of the devices acting as a relay may significantly improve the network performance. This emerging paradigm, known as indirect network connection mode, represents a killer feature of the upcoming 5G technology to counteract the negative effects of the channel propagation conditions at the network edges, by allowing D2D communications directly between devices if the line of sight (LOS) signals are not available in a reliable manner. The advantages deriving by the D2D communications are appreciable in terms of (i) improvements on offloading flexibility; (ii) reduction of the discovery procedures cost for the context-aware services and applications; (iii) improvements in resource utilization; (iv) latency and energy consumption reduction [6; 7].  
Therefore, in 5G environments, the proper relays selection plays an essential role for designing networks able of energy efficient short-range D2D communications [8, 9].  
The paper investigates the problem of multicast communications considering relay-based schemes operating in the decoding-and-foreword mode and D2D communications to share a same content simultaneously from a source to a target users community according to the multicast paradigm. In particular, we focus here on a relays selection policy based on the the joint use of devices reputation information [10, 11] and link propagation conditions.  
Generally speaking, the reputation coefficient may assume a wide range of meanings, from the hardware capabilities such as the battery life time, to the security level, to name a few. Therefore, with the aim at proposing a general purpose relay selection framework, we have left unspecified the meaning of the reputation coefficient in what follows.  
This paper focuses on optimizing the content delivery delay (defined as the minimum time period in which all the devices retrieve successfully the interesting content) and the number of the involved relays with the aim at minimizing the overall energy consumption, by adopting a matching theory (MT) approach and supposing one shared channel. Therefore, the main contributions of this paper are summarized as follows  
* We propose a relay-based content delivery approach to offload the data traffic through D2D communications, by establishing energy efficient connections. In particular, multiple relays can be selected among all the interested users to spread the content and minimize the end-to-end delivery delay.
* The adopted matching process achieves the combinations of relays and destinations according to a reputation strategy and the propagation conditions of the links from source to relays and the links from relays to end-users with respect to the direct link (i.e., from source to end-users). This joint process minimizes the worst delivery delay and the number of the involved relays, by resorting to the capacitated house allocation matching problem.  
The rest of paper is organized as follows. In Section 2 and in-depth literature review is provided. In Section 3, the system model is detailed and the proposed strategy is presented in Section 4. In Section 5, the experimental results are shown. Finally, the conclusions are drawn in Section 6.