## 1 Introduction
### Contribution  
Despite the advancements in the existing literature, two critical challenges remain inadequately addressed. Firstly, existing BH technology resource allocation methods primarily focus on beam assignment within a single time slot, lacking global optimization across the entire BH cycle. This limits the effective utilization of temporal resources within the system. Secondly, while these studies have provided important theoretical and algorithmic foundations, most either target a single resource dimension or rely on fully coupled optimization models that impose high computational complexity. Resource allocation in BH-based systems involves multiple coupled dimensions--beam-footprint partitioning, time-slot scheduling, and frequency-power allocation. The resulting joint optimization problem is high-dimensional and computationally complex. Therefore, it is essential to develop a scheduling framework that can effectively decouple spatio-temporal-spectral resources and adopt a hierarchical optimization strategy to achieve near-optimal system performance with reduced computational complexity. To address the challenges of high-dimensional coupling and suboptimal temporal utilization in BH-based satellite systems, this paper investigates a BH-based beam scheduling and resource allocation framework. In contrast to conventional approaches that rely on joint iterative optimization, the proposed framework adopts a progressive, dimension-wise optimization strategy, in which each resource dimension is handled independently in a sequential manner, without requiring cross-iteration between modules. This enables efficient and scalable scheduling with significantly reduced computational complexity. Specifically, we develop a spatio-temporal-frequency resource allocation framework tailored for heterogeneous GEO-LEO satellite networks. The objective is to fully exploit the time, spatial, and spectral resources available to the LEO system, thereby enhancing the overall system throughput. Within the proposed framework, beam-footprint partitioning,time-slot assignment, beam-level power allocation, and user-specific frequency-power allocation are jointly addressed over the entire BH cycle. Each component is independently modeled and solved using modular algorithms, including clustering and integer linear programming (ILP) for spatial optimization, a hybrid greedy-simulated annealing approach for temporal scheduling, and a Hungarian-water-filling-convex optimization pipeline for spectral and power domains. Although the resource dimensions are optimized sequentially, cross-domain dependencies are minimized and there is no need for iterative feedback between stages, thereby enabling tractable and flexible resource coordination under realistic constraints. The main contributions of this paper are summarized as follows:  
* Beam-Footprint Partitioning: to improve the coverage efficiency of multi-beam LEO satellite systems, we develop a multi-phase beam partitioning strategy that integrates density-aware clustering with ILP. The initial beam layout is guided by user distribution, where regions of high user density are prioritized during clustering. Subsequently, ILP is applied to optimize the association between users and beams, aiming to adjust beam center positions so that users are located as close as possible to beam centers for enhanced channel gain. An iterative adjustment step eliminates low-efficiency beams and reallocates their users, thereby increasing beam utilization efficiency and enabling better adaptation to spatially non-uniform traffic patterns. * Time-Slot Beam Scheduling: for scheduling beam positions across time slots within a BH cycle, we develop a hybrid approach, combining a greedy algorithm and simulated annealing. A feasible initial schedule is generated using a greedy filling strategy under constraints on minimum beam separation and per-slot beam limits.