## 2 Related Works  
Recently, several efforts have been made in providing schemes to efficiently perform D2D communications, especially in 5G networks. Some examples are represented by papers [1, 5, 12], that detail the challenges of the D2D communications contextualized to the 5G cellular networks. The problem of the relay selection for the point-to-point communications in the absence of the LOS has been investigated in [13, 14, 15].  
The improvements in outage probability and capacity deriving by the introduction of the nodes acting as relays are analyzed in paper [16], showing the close forms for the ergodic multicast capacity and the outage probability. The optimal joint problem of the relay selection and the channel assignment is investigated in paper [17], by using a lexicographic max-min approach. The relay selection issue has been addressed also in paper [18], in which a distributed scheme is realized on the basis of a maximum distance threshold, involving the exchange of local tables to select the suitable relays. Roughly speaking, these relay selection policies choose the best propagation link analyzing the instantaneous channel state information in a network assisted vision.  
A game theory-based approach has been shown in paper [19], in which a Stackelberg game has been formulated in order to optimize both the interference management and the D2D offloading, taking into account also the monetary aspect. Differently, the social behavior of users, which frequently use on-line social applications, such as Facebook, Twitter, YouTube, Instagram, and so on, has suggested the establishment of D2D groups in which each participant has the same interest in content sharing to drive cooperative D2D/relaying communications design [20, 21]. The maximization of the amount of data relayed through D2D communications and the stability of the D2D transmissions are the main target of paper [22] for social networks. Similarly, the communications stability is taken into account even in [23], where large scale machine-to-machine (M2M) networks are well investigated. The authors in [23] pose emphasis on the fact that the social awareness during relay selection positively impacts on the network performance, providing improvements also in reference to the stability of communications. The problem of the throughput maximization of the network is studied in paper [24]. In this case, the relay selection scheme considers both the distance between the relay and the source-destination pair, and the corresponding social trust coefficient. The discovery procedure and the resource allocation problem contextualized to the content delivery issue is studied in [25], showing a two-sided preferences matching approach, considering both the social connections and the physical information, aiming at maximizing the sum rate for the D2D communities. A matching approach has been adopted also in paper [26], where the main objective was the maximization of the energy efficiency by keeping high levels of quality of service. Similarly to the work [25], our paper considers the same problem, i.e., the content delivery to end-users through relays. Differently, we focus our analysis on the of end-to-end delivery delay problem for multicast communications. Therefore, a relay selection is proposed, where each device can act as a relay by resorting to a one-sided preferences matching game, formulated as a capacitated house allocation matching problem [27]. Despite several papers proposing the matching theory as a solution as in [28], the problem of multicast delivery has not been taken into account.