## 1 Introduction  
At present, the global commercialization of 5G mobile communication networks is advancing [1, 2, 3, 4, 5] ; however, due to the constraints of geographic conditions and construction costs, it is impossible to achieve seamless global coverage [6]. 6G non-terrestrialnetwork (NTN) technology [7], which is the convergence of satellite communication networks and 5G, has gradually become the focus of attention in academia and industry [8; 9; 10; 11]. For example, the 3 rd Generation Partnership Project (3 GPP) proposed four start-earth converged network architecture schemes based on on-planet processing/transparent forwarding with/without relaying [12], and ITU proposed four application scenarios for the convergence of 5G and satellite communications [13]. Compared to medium earth orbit (MEO) and geostationary earth orbit (GEO) satellite networks, LEO-MSN has the advantages of low transmission delay, low propagation loss, high regional capacity, and low manufacturing and launching costs, and it has been considered to be an essential part of next-generation communication networks [14; 15; 16; 17; 18]. In recent years, a large number of low-orbit mega-constellation programs, represented by StarLink, OneWeb, and Kuiper, have been proposed. These constellations usually consist of hundreds or thousands of satellites and operate in 200-2000 km near-Earth orbits at speed of 7.2 km/s [19]. Among them, SpaceX's StarLink constellation has grown the most rapidly, with more than 6300 satellites deployed as of June 2024 [20], providing satellite-accessed internet communication services to more than 2.5 million subscribers in more than 70 countries and territories, with up to 500 Mbps downlink speed, 40 Mbps uplink speed, and 20 ms latency internet access services [21]. As one of the key technologies for LEO-MSN communication, routing significantly impacts traffic transmission delay and quality of service [22]. However, in LEO-MSN, the routing technology still faces the following challenges. Firstly, the massive scale of mega constellations and the nodes with high dynamics make the network topology highly complex, and it is difficult to construct stable traffic transmission paths. In addition, the uneven distribution of the traffic in time and space makes the LEO-MSN face problems such as load imbalance and congestion [23]. With the increasing capacity of on-board load processing, the LEO-MSN network architecture has evolved from transparent payload (TP) mode to regenerative payload (RP) [24] mode. According to the Service-Based Architecture (SBA) proposed by 3 GPP [25], the functions of the access network, transmission network, and core network can be segmented on demand and deployed flexibly on the satellite or the ground, which brings new constraints to routing and transmission strategies. Under the new service architecture, the routing policy must satisfy the Service Function Chain (SFC) constraints [26], i.e., the traffic transmission path from the source node to the target node must pass through the nodes of relevant service functions in order. In the dynamic topology and resource-constrained environment of LEO-MSN, the introduction of SFC constraints will make satellite routing face more challenges. First, the functional nodes need to centrally process multi-path traffic, resulting in drastic fluctuations in network load, which becomes a key bottleneck for network congestion. In addition, the high-speed movement of satellites causes frequent reconstruction of SFC paths. Traditional routing methods lack global perception of the functional chain status and are difficult to maintain service continuity during link switching, further exacerbating the risk of transmission interruption. Facing the new large-scale satellite network scenarios and the introduction of SFC constraints, existing satellite routing methods are no longer applicable, which is mainly due to the fact that some of the methods only optimize from a single metric. For example, refs.