## 1 Introduction
Additionally, some articles have focused on the software-defined networking and network function virtualization in satellite networks, such as [21; 22; 23]. To address high dynamics in LEO satellite networks, [PERSON] et al. [24] proposed a time-expanded graph-based model, while [PERSON] et al. [25] developed a model for satellite network topology that considered routing and satellite mobility. [PERSON] et al. [2] defined a task queue and adopted a multi-hop model to represent the transmission process of ISLs. To achieve a more accurate analysis of SEC characteristics, we developed a mixed-integer linear programming (MILP) model based on [1; 2; 25] that also considers satellite orbit elements and data dependencies between tasks. Figure 1: Task offloading in an SEC network with data-dependent constraints. The existing approaches for solving task offloading in SEC can be categorized into two categories: centralized and distributed. Centralized algorithms include heuristics [26], meta-heuristics [15; 27; 28], game algorithms [29], optimization methods [30], and reinforcement learning [3; 31], all of which have demonstrated their effectiveness in various application scenarios. In contrast, distributed approaches, such as dynamic group learning distributed particle swarm optimization [32] and the multi-agent actor-critic reinforcement learning algorithm [33], cannot guarantee optimal results. As metaheuristic approaches can efficiently handle large-scale problems within polynomial time [34], we propose an MOWPS algorithm to address this issue. To avoid unexpected computation delays and unsuccessful application execution, it is essential to consider the dependency constraints of tasks in SEC [26]. [PERSON] et al. [28] proposed two offloading schemes (parallel and sequential) to address task dependencies, while [PERSON] et al. [27] introduced a queue-based method for task offloading based on their allowable execution times. [PERSON] et al. [3] modeled tasks with dependencies as directed acyclic graphs, then proposed an attention mechanism and proximal policy optimization collaborative algorithm to obtain the best offloading strategy. [PERSON] et al. [35] proposed a hybrid genetic binary particle swarm optimization algorithm in which the task sequence is determined by the depth-first algorithm. [PERSON] et al. [26] and [PERSON] et al. [15] both used heuristic algorithms to generate task sequences, and [PERSON] et al. [36] proposed a ready queue approach for dynamic applications. However, obtaining the execution status of each satellite at any time in satellite networks can be costly. As a graphical and mathematical modeling tool [37; 38], a Petri net [39] can effectively describe the coupling between applications, tasks, and edge servers. However, no studies have used Petri nets to model task offloading in SEC. In this paper, we first present a Petri-net-based constraint amending method to handle dependency constraints. Additionally, task offloading in SEC involves multiple optimization objectives. When there are three or more objectives, the existing algorithms face challenges. First and foremost, as the number of objectives increases, almost all solutions in population become nondominated [40], making Pareto rank-based methods like NSGA-II invalid [41]. Secondly, solutions in high-dimensional space are typically sparsely distributed in the objective space [42], which makes it harder to maintain diversity. Finally, some metrics such as hypervolume may incur significant computational overhead. Prior work has been performed by a few scholars. For example, [PERSON] et al. [27] used Pareto-optimal relations to obtain an archive set and introduced a grid method to maintain diversity. [PERSON] et al. [15] proposed a multi-swarm co-evolutionary mechanism in which each population focuses on different objectives and subsequently performs collaborative optimization. [PERSON] et al. [43] devised a two-stage multi-objective optimization approach aimed at striking a balance between transmission rate and power consumption. [PERSON] et al.