This paper studies a network with multiple unmanned aerial vehicle (UAV) relays assisting the wireless communication between a base station (BS) and ground users (GUs). All the UAV relays hover at a fixed altitude and each provides connections to a group of GUs. For the first time, the radio-map-based design is considered in UAV-assisted relaying networks, where the radio maps are based on the geographic layout in reality and presenting the realistic channel pathloss. By jointly designing the UAV placement and user assignment, we aim at maximizing the minimum throughput among all GUs in the network. As the radio maps rely on the real layout in the environment, the pathloss is neither convex nor continuous in the location (coordinates), which makes the related designs be highly difficult to analyze. Thus, we propose a particle swarm optimization (PSO)-based design for the problem. We first derive the optimal user assignment scheme and then derive an equivalent reformulation of the problem. Finally, the UAV placement is optimized by deploying PSO technologies, providing an efficient system performance. Via simulation, the performance advantages of the proposed design are confirmed.