# An Accelerated Maximum Flow Algorithm with Prediction Enhancement in Dynamic LEO Networks
## 3 System Model and Problem Formulation
### Definitions and Problem Formulation  
Considering a LEO satellite network comprises a source satellite \(s_{1}\), a set of relay satellites \(\{r_{1},r_{2},\ldots,r_{N}\}\), and a ground station \(g_{1}\), the primary objective is to maximize the flow of spatial data from the source to the ground station within a specified time horizon \(T=[t_{0},t_{k})\). Data transmission can occur directly or via relay satellites, with operations constrained by the storage capacity of each satellite, the time-varying transmission capacity of communication links, and the energy budget available to each satellite.  
The time horizon \(T\) is divided into \(k\) discrete time slots, denoted as \(\{\tau_{1},\tau_{2},\ldots,\tau_{k}\}\), where each time slot \(\tau_{h}=[t_{h-1},t_{h})\) represents a static snapshot of the network. Satellites in the network are equipped with limited storage capacity, which restricts the amount of data that can be buffered at any given time. Formally, the total data stored at a satellite \(v_{i}\) during the time horizon must not exceed its buffer capacity \(S_{v_{i}}\). It ensures that the satellite can manage incoming and outgoing data efficiently and reflects the operational limitations inherent in satellite hardware.  
Communication between satellites and the ground station relies on directed communication links with time-dependent transmission capacities. For a link between two nodes, represented as \((v_{i},v_{j})\), the volume of data transmitted during a time slot \(\tau_{h}\) cannot exceed the capacity \(\mathcal{C}_{v_{i},v_{j}}^{h}\). These capacities vary across time slots due to the dynamic nature of LEO networks, which experience frequent topology changes as satellites move along their orbits.  
Energy is another critical resource in LEO satellite networks. Each satellite has a finite energy budget, denoted by \(E_{v_{i}}\), which governs its ability to transmit and process data throughout the time horizon. As satellites consume energy to maintain communication links, buffer data, and relay information, the total energy consumed by a satellite over all time slots must remain within its budget.  
The goal of the network is to maximize the total data flow \(F\) from the source \(s_{1}\) to the ground station \(g_{1}\) while satisfying the constraints imposed by storage, transmission capacity, and satellite energy. The total flow \(F\) is calculated as the sum of all data transmittedfrom \(s_{1}\) to \(g_{1}\) across all time slots. Formally, the optimization problem can be expressed as follows:  
\[F=\sum_{h=1}^{k}\sum_{(s_{1},h)\in\mathcal{E}}f_{s_{1},h}^{h} \tag{1}\]  
where \(f_{s_{1},h}^{h}\) represents the data flow from \(s_{1}\) to its neighboring node \(\mu\) during the time slot \(\tau_{h}\). The flow must respect the constraints associated with storage, transmission capacity, and energy, ensuring the solution remains feasible under the physical and operational limitations of the network, as formulated below:  
\[0\leq f_{\tau_{1},v_{j}}^{h}\leq C_{\tau_{1},v_{j}}^{h},\quad\forall i,j\in[1, N],\forall h\in[1,k] \tag{2}\]  
\[\sum_{h=1}^{k}\sum_{j:(v_{j},v_{j})\in\mathcal{E}}f_{v_{j},v_{j}}^{h}\leq S_{ v_{j}},\quad\forall i\in[1,N] \tag{3}\]  
\[\sum_{h=1}^{k}E_{v_{j}}^{h}\leq E_{v_{j}},\quad\forall i\in[1,N] \tag{4}\]  
To align with practical satellite operations, we consider a variety of constraints that limit the network's ability to handle data flow in Sections 3.2 and 3.3.