In this article, we establish a novel framework for a multi-user mobile edge computing (MEC) network in which a set of users with high downlink rate demands and a set of users with intensive computation tasks can collaborate to achieve a mutually-beneficial scenario such that completion time of the tasks is reduced and the base station (BS) can send more information at a higher rate to the downlink users. Specifically, by leveraging non-orthogonal multiple access (NOMA) for uplink and downlink traffic, the user with the computation task can offload shares of the computation task to the edge cloud and the downlink user. At the same time, this user forwards the information it receives from the BS to the downlink user. In this set up, we jointly optimize the communication resources, computational resources at the edge cloud and user devices, pairings among the two sets of users, the shares of computation tasks, and relay bits to minimize the total task completion time while satisfying downlink users’ incentive requirements. For a network with a single computation demanding user and a single downlink user, the optimal solution to the problem is provided. For a network with multiple users, the problem is non-convex and computationally challenging. Hence, we propose an efficient, low complexity algorithm that utilizes the bottleneck matching algorithm, convex optimization, and the block coordinate descent scheme to obtain a locally-optimal solution. Simulation results demonstrate that, as compared with the state-of-the-art, the total task completion time is greatly reduced (32%–51%), and a large computational energy savings at the edge cloud (38%–55%) is achieved. Simultaneously, the downlink users’ rates improve compared to the orthogonal transmission.