## 4 Algorithmic Approaches
While the algorithm found solutions that are very close to those found by model \(M_{2}\), the CPU processing times are much lower than those required by \(M_{2}\) and are comparatively more efficient than in the case of the benchmark networks for the 20% density scenario, as shown in Figure 5.

For the 30% controller density scenario, the comparison of solutions found is even closer. Note that in Figure 6, the scale of the Y-axis on the left is smaller, and although

Figure 4: This figure represents both the objective function values on the left and the CPU times on the right side for the benchmark networks studied at a 40% controller density. Figure 5: This figure represents both the objective function values on the left and the CPU times on the right for the random networks at a 20% network density. the CPU times are better than \(M_{2}\), compared to the previous scenario, the performance is slightly lower. Finally, in the 40% controller density scenario, the trend of closeness in the solutions found by the algorithm compared to \(M_{2}\) continues, and this closeness becomes stronger for larger networks. Similarly, the processing time used by the algorithm is much lower than that required by \(M_{2}\) for the majority of the studied networks, with the advantage being clearer for larger networks. From these analyses, including the trends presented in Figure 7, it is suggested that the proposed algorithm can be used for larger networks. Based on this final suggestion, a last experiment was conducted with larger random networks, as can be seen in Table 5 bellow. Figure 6: This figure represents both the objective function values on the left and the CPU times on the right for the random networks at a 30% network density. Table 3.0 This figure represents both the objective solutions on the left and the CPU times on the right for the random networks at a 40% network density. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Network Name**} & \multicolumn{4}{c}{**20\% Controllers Density**} & \multicolumn{4}{c}{**30\% Controllers Density**} & \multicolumn{4}{c}{**40\% Controllers Density**} \\ \cline{2-13}  & \multicolumn{2}{c}{\(M_{2}\)} & \multicolumn{2}{c}{**Algorithm**} & \multicolumn{2}{c}{\(M_{2}\)} & \multicolumn{2}{c}{**Algorithm**} & \multicolumn{2}{c}{\(M_{2}\)} & \multicolumn{2}{c}{**Algorithm**} \\ \cline{2-13}  & **Solution** & **CPU** & **Solution** & **CPU** & **Solution** & **CPU** & **Solution** & **CPU** & **Solution** & **CPU** \\ \hline Random (150) & 0.029 & 3600 & 0.044 & 24.1 & 0.009 & 3600.1 & 0.014 & 37.7 & 0.004 & 3600.1 & 0.006 & 23.1 \\ Random (200) & 0.019 & 3600.1 & 0.029 & 41.9 & 0.006 & 3600.1 & 0.009 & 50.8 & 0.002 & 3600.1 & 0.004 & 41 \\ Random (300) & 0.01 & 3600.1 & 0.016 & 95.3 & 0.003 & 3600.1 & 0.005 & 106.4 & 0.001 & 3600.3 & 0.002 & 93.4 \\ Random (400) & 0.007 & 3600.1 & 0.011 & 168.6 & 0.002 & 3600.2 & 0.003 & 171.3 & 0.001 & 3600.1 & 0.001 & 164.7 \\ Random (500) & 0.006 & 3600.2 & 0.008 & 263.8 & 0.002 & 3600.2 & 0.002 & 270.5 & 0.001 & 3600.3 & 0.001 & 261.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Analysis of Algorithm 1 versus model \(M_{2}\) for larger random networks. From Table 5, it can be observed that the behaviors for all scenarios are similar to the above tables. In the scenario with a 40% controller density, the results show reduced CPU processing times. Additionally, upon analyzing each scenario and comparing the results obtained with \(M_{2}\) and the proposed algorithm, it can be noted that as the size of the network increases, the difference between the solution of \(M_{2}\) and the proposed algorithm decreases. Furthermore, there is a significant reduction in CPU processing time compared to model \(M_{2}\), with an average between 1% and 7% of the CPU time required by model \(M_{2}\). The results obtained from analyzing each scenario lead to the conclusion that the use of Algorithm 1 is recommended for larger networks again.