## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.6 Problem Modeling and Optimization Algorithm Design
Another situation arises when constraints are active, meaning that they are satisfied as equalities, i.e., \(g_{i}(x)=0\), and the Lagrange multipliers associated with these constraints are strictly positive, i.e., \(\lambda_{i}>0\). This occurs when resource limits, such as bandwidth or computational capacity, are fully utilized. In this case, the optimization adjusts resource allocation among the nodes to ensure that all active constraints are satisfied, while maximizing the overall performance. To handle this scenario, we can consider fine-tuning the Lagrange multipliers, allowing the optimization process to strike a balance between satisfying the constraints and maximizing system efficiency. A third scenario arises when the optimization problem becomes infeasible due to conflicting constraints. This situation is characterized by \(g_{i}(x)>0\) for all \(i\), where no feasible solution can simultaneously satisfy all constraints. This may happen when the required resources exceed the available capacity, such as when the demand for bandwidth or power exceeds the network's capability. To address this, we can introduce slack variables that relax some of the constraints slightly, allowing minor violations. Additionally, we can incorporate a penalty function into the objective function to penalize these constraint violations. This approach would allow the optimization to converge to a feasible solution, even when the problem initially seems unsolvable, ensuring robustness in resource-tight conditions. To achieve optimal resource allocation, the CMPT framework employs an algorithm that integrates Shapley-value-based resource distribution with Lagrangian multiplier optimization techniques. The detailed implementation of this optimized resource allocation process is presented in Algorithm 4. ```
1:Input: \(\lambda_{l,f}\), \(\mu_{l,m,f,t}\), \(\
u_{l,m,f,t}\),



for each node. For each participant \(i\in N\), the Shapley value is first calculated (Lines 1-2), followed by resource allocation based on the computed Shapley value (Line 3). After resource allocation for all participants is completed, the Lagrangian function is defined (Line 5), and the corresponding dual problem and KKT conditions are solved (Lines 6-7). Finally, the optimized resource allocation results \(P_{l,m,f,t}\), \(B_{i}\), and \(P_{i}\) are returned (Line 8). ```
1:Input:\(U=\{U_{1},U_{2},\ldots,U_{N}\}\), link reliabilities \(\mu_{ij}^{E}\), path reliabilities \(R_{h}\), coded packets \(Y_{i}\)
2:Output: Optimal resource allocation \(P_{l,m,f,t}\), bandwidth allocation \(B_{i}\), power distribution for each node \(P_{i}\). 3:for each participant \(i\in N\)do
4: Calculate Shapley value as Equation (43);
5: Distribute resources based on \(\phi_{i}(v)\);
6:endfor
7: Define Lagrangian as Equation (46);
8: Solve dual problem as Equation (47);
9: Solve KKT conditions as Equation (48);
10:Return: Final resource allocation \(P_{l,m,f,t}\), \(B_{i}\), \(P_{i}\). ```

**Algorithm 4** Cooperative Resource Allocation with Shapley Value and Dual Optimization (CRA)

This Algorithm 5 first receives the satellite node set \(N\), task set \(T\), network topology \(G(V,E)\), and resource capacity \(R\) as input (Line 1), and then aims to output the task allocation \(A\), power allocation \(P\), and bandwidth allocation \(B\) (Line 2). It begins by preprocessing the network information and task requirements to compute the utility values \(U\), link reliability matrix \(\mu^{E}\), path reliability \(R^{h}\), and the encoded data packets \(Y\) necessary for subsequent resource allocation (Line 2). The algorithm then performs domain clustering based on the network topology and clustering criteria, to divide all nodes into multiple sub-domains \(D\) (Line 4). For each domain \(D_{k}\) in \(D\), it approximates the local Shapley value \(\phi_{i}\) to evaluate the fair contribution of each node within the domain, significantly reducing the computational complexity compared to global Shapley calculation (Lines 5-7).