# An Accelerated Maximum Flow Algorithm with Prediction Enhancement in Dynamic LEO Networks
## 2 Related Work
### Advancements in Maximum Flow Algorithms  
The Ford-Fulkerson method [7] is a classical approach to computing maximum flow in networks. Variants of the maximum flow algorithm have introduced path-finding optimizations to improve runtime. For example, the Edmonds-Karp algorithm [16] uses breadth-first search (BFS) to identify the shortest augmenting path, while [PERSON]'s algorithm [8] employs a layered network structure to accelerate convergence in certain types of networks. These methods have been widely used in various applications, achieving practical efficiency in static networks.  
Recently, learning-augmented techniques have emerged as a promising direction for improving runtime by leveraging prior knowledge. For instance, [PERSON] et al. [9] proposed a learning-augmented maximum flow algorithm that uses predicted flow values to accelerate computation. Their method achieves a runtime of \(\mathcal{O}(m\eta)\), where \(m\) is the number of edges and \(\eta\) is the \(l_{1}\) prediction error. However, these predictions do not guarantee adherence to edge capacity constraints, limiting their direct applicability to certain network problems.  
While other works, such as [17], addressed specific cases like 0-1 min-cost flow, their techniques rely on heavily constrained problem setups (e.g., binary capacities) and do not generalize to the broader maximum flow problem. Similarly, frameworks for warm-starting algorithms [18] are limited to specific problem types, such as L-convex optimization, and are not directly applicable to flow networks. Ref. [19] demonstrated the use of predictions to enhance the runtime of generalized sorting.  
Additionally, research on leveraging data-driven approaches for algorithm optimization has gained traction. For instance, a study presented in [20] explored the integration of machine learning predictions within traditional algorithmic frameworks, with a focus on improving efficiency. Furthermore, ref. [21] discussed how predictive models can be utilized to inform and accelerate algorithmic processes, particularly in the domain of network flow problems. In a related area, data-driven algorithm design, e.g., [22; 23], has emerged as a promising direction for leveraging data to optimize algorithm performance, further highlighting the potential of incorporating machine learning into traditional algorithmic paradigms.  
Beyond these advancements, recent research efforts have focused on enhancing the efficiency and versatility of maximum flow algorithms within dynamic and constrained environments. Notably, [PERSON] [24] and [PERSON] et al. [25] developed fast algorithms for computing approximate maximum flows, offering substantial speed improvements for large-scale networks. Furthermore, [PERSON] et al. [26] introduced an algorithm that computes the maximum flow in nearly linear time. However, it is not yet feasible for practical applications in the real world due to its restrictive data assumptions, complex implementation, and inability to fully meet the demands of high real-time and highly dynamic scenarios. [PERSON] and [PERSON] [27] empirically demonstrated the potential to accelerate the push-relabel algorithm by employing warm-starts on similar networks, underscoring the opportunity to leverage prior solutions for faster computations.  
While TEG and e-TEG have laid a solid foundation for modeling dynamic satellite networks, existing algorithms for solving maximum flow problems in these models do not fully leverage the potential of predictive acceleration. Current warm-starting techniques, though promising, either focus on restricted problem setups or lack integration with dynamic, capacity-constrained environments like e-TEG.  
To provide a clearer understanding of the distinctions and connections among these models and algorithms, Table 1 provides a systematic comparison of the TEG, e-TEG models, and representative maximum flow algorithms in terms of their characteristics, advantages, and limitations.