## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.6 Problem Modeling and Optimization Algorithm Design
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Datasets**} & \multirow{2}{*}{**Average**} \\ \cline{2-2} \cline{4-6}  & **D1** & & **D2** & & **D3** & **D4** \\ \hline
**CMPT** & **30.15454** & **28.36954** & **27.38451** & **22.35846** & **27.56626** \\ LRM [44] & 13.47757/123.87\% & 10.23800/177.15\% & 12.88180/112.59\% & 12.84011/74.08\% & 12.35987/121.41\% \\ AHPSO [43] & 27.10558/11.26\% & 7.96670/256.03\% & 16.48721/66.09\% & 16.38320/36.47\% & 16.48567/67.27\% \\ AABC [42] & 15.28467/97.37\% & 10.69852/165.22\% & 10.88180/151.65\% & 12.84011/74.08\% & 12.42628/121.79\% \\ AACO [41] & 27.13852/11.12\% & 19.56770/45.03\% & 16.48721/66.09\% & 16.38320/36.47\% & 19.39466/42.12\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Objective function values and optimization rates across four datasets. Figure 6: Task execution time between algorithms across four datasets. Figure 7: Growth rate of task execution time across four datasets. As shown in Figure 8, CMPT demonstrated sustained advantages in task execution time across the different time steps. From \(t_{1}\) to \(t_{10}\), CMPT consistently outperformed the other algorithms, maintaining significantly lower execution times and showcasing superior real-time performance and efficiency. This is primarily attributed to its optimized multi-path scheduling strategy and real-time feedback mechanism, which dynamically adjust task paths to minimize resource waste and link latency. In contrast, AABC and LRM exhibited persistently higher task execution times throughout the time steps, further indicating their inefficiency in handling real-time scheduling tasks. ### Summary

The experimental results demonstrate that the Coding Multi-Path Theory (CMPT) method achieved high throughput efficiency, balanced resource utilization, and superior real-time performance in multi-satellite collaborative computing and dynamic network environments. Specifically, across the four datasets with varying numbers of satellites, CMPT attained an average objective function value of 27.56626, significantly outperforming baseline algorithms such as LRM (12.35987), AHPSO (16.48567), AABC (12.42628), and AACO (19.39466). Moreover, CMPT exhibited outstanding scalability and stability for task execution time, with an average execution time of 271.91026 units, markedly better than LRM (1743.62500), AHPSO (710.29763), AABC (652.25000), and AACO (331.35843). This superior performance is attributed to CMPT's integration of Nash bargaining game and Lagrangian dual optimization, which ensures coordinated resource allocation and efficient load balancing, even under multiple interference constraints and large-scale task loads. However, despite its advantages, CMPT also presents certain limitations. For instance, the algorithm's computational complexity may increase with extremely large-scale networks or highly dynamic environments, potentially impacting its real-time responsiveness. Additionally, the reliance on accurate real-time network state monitoring necessitates ro

Figure 8: Task execution time across discrete time steps. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Datasets**} & \multicolumn{1}{c}{**Average**} \\ \cline{2-6}  & **D1** & **D2** & **D3** & **D4** & \\ \hline
**CMPT** & **54.68548** & **26.54845** & **348.25484** & **658.15428** & **271.91026** \\ LRM [44] & 154.50000\(/-\)64.60\% & 212.00000\(/-\)87.48\% & 2846.50000\(/-\)87.77\% & 5759.50000\(/-\)88.57\% & 1743.62500\(/-\)84.40\% \\ AHPSO [43] & 45.96000\(/\)19.00\% & 144.25214\(/-\)81.60\% & 894.65125\(/-\)61.07\% & 1756.32514\(/-\)62.53\% & 710.29763\(/-\)61.71\% \\ AABC [42] & 45.00000\(/\)21.52\% & 125.00000\(/-\)78.77\% & 485.00000\(/-\)28.16\% & 1956.00000\(/-\)66.36\% & 652.25000\(/-\)34.38\% \\ AACO [41] & 37.55238\(/\)45.62\% & 11.42547\(/\)132.37\% & 430.80934\(/-\)19.20\% & 847.64851\(/-\)22.38\% & 331.35843\(/-\)18.03\% \\ \hline \hline \end{tabular}
\end{table}