## 2 Related Work  
In order to improve spectrum efficiency in satellite communication systems, a large number of radio resource allocation algorithms have been proposed. In [14], the total throughput of all cells is maximized under the condition that system throughput and QoS requirements are considered. In [15], in the user-intensive scenario, satellite and ground networks cooperate to complete the backhaul transmission and ensure the QoS of traffic. [PERSON], [PERSON] et al. [16] consider a variety of scenarios and optimizes the data rate and resource allocation of all nodes. However, these resource allocation algorithms are mainly applied to single-beam satellite communication systems or terrestrial heterogeneous networks, which cannot meet the demands of radio resources allocation of spectrum multiplexing between different satellites in multi-beam satellite communication systems. Among them, the works of resource allocation in multi-beam satellite communication systems have been studied before. In [17], a congestion pricing game model is proposed to meet the different demands of ground users in different satellite links in non-geosynchronous satellite systems. In [18], a dynamic bargaining resource allocation algorithm is proposed, and the numerical test results show the influence of different algorithm parameters on the optimal pricing and the profit of the satellite system. In [19], a perfect market incentive-based technology for spectrum optimization of GEO multi-beam satellite systems with different user preferences to improve spectrum utilization is designed. The current multi-beam satellite cooperative transmission systems only consider the resource allocation between the inner beams of a single satellite. For multi-beam satellite communication systems, only the resource allocation of CDMA (Code Division Multiple Access) is considered between different satellites and different beams [20; 21]. In [22], they introduce a potential game-based approach to implement collaborative user scheduling and power allocation in the uplink multi-beam satellite IoT (S-IoT) networks. In this paper, the problem of radio resource allocation based on OFDM is studied in multi-beam satellite communication systems. Resource allocation of macrocell and microcell cooperative transmission in terrestrial heterogeneous networks is being studied intensively. In [23; 24; 25; 26], resource allocation of macrocell and microcell cooperative transmission in terrestrial heterogeneous networks is introduced. Terrestrial communication systems mainly make use of the different coverage range and transmitting power of macrocell and microcell to carry out cooperative transmission, but the priority of the satellite coverage is not different. There is little difference in transmission power. If power and coverage are used for user division, it is easy to cause misdivision, resulting in lower transmission efficiency. In order to realize the spectrum multiplexing, effective radio resource allocation is the key to improve the system performance. Heuristic algorithms for radio resource allocation have been tried before. In [27], a resource management scheme aiming at maximizing resource utilization and user service weight is proposed, and an improved cuckoo algorithm for optimization to ensure QoS of high-priority users is introduced. In [28], a low-orbit satellite communication system based on wave hopping is designed, and the greedy algorithm to solve the resource allocation of the system is used, which improves the system throughput and space-borne resource utilization. However, the traditional heuristic algorithms can only find the local optimal solution. It is difficult for the traditional heuristic algorithm to solve the problems of high dynamic multi-dimensional resources optimization and Time-division and discontinuous packet segmentation in multi-beam satellite communication systems. In [29; 30], the subcarrier resource allocation based on the OFDM system in the visible light terrestrial system is studied.