## 4 Algorithmic Approaches
For the 30% controller density scenario, it can be observed that both models achieve optimal results, with the random networks

\begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Network Name**} & \multicolumn{6}{c}{\(M_{1}\) with 20\% Controllers Density} & \multicolumn{6}{c}{\(M_{2}\) with 20\% Controllers Density} \\ \cline{2-11}  & **Solution** & **Best Bound** & **MipCap** & **CPU** & **NC** & _B\& **R Nodes** & **Solution** & **Best Bound** & **MipGap** & **CPU** & **NC** & _B\& **Nodes** \\ \hline Random (20) & 0.44 & 0.44 & 0 & 2.55 & 4 & 71 & 0.44 & 0.44 & 0 & 0.6 & 4 & 1 \\ Random (25) & 0.38 & 0.38 & 0 & 6.7 & 5 & 366 & 0.38 & 0.38 & 0 & 3.18 & 5 & 1 \\ Random (30) & 0.32 & 0.32 & 0 & 38.4 & 6 & 1879 & 0.32 & 0.32 & 0 & 3.74 & 6 & 1 \\ Random (35) & 0.29 & 0.29 & 0 & 112.89 & 7 & 1282 & 0.29 & 0 & 36.67 & 7 & 956 \\ Random (40) & 0.26 & 0.26 & 0 & 220.94 & 8 & 1030 & 0.26 & 0.26 & 0 & 191.11 & 8 & 1039 \\ Random (45) & 0.23 & 0.23 & 0 & 394.44 & 9 & 1114 & 0.23 & 0.23 & 0 & 61.52 & 9 & 888 \\ Random (50) & 0.12 & 0.12 & 0 & 669.12 & 10 & 1159 & 0.12 & 0.12 & 0 & 180.75 & 10 & 998 \\ Random (55) & 0.19 & 0.19 & 0 & 1205.55 & 11 & 2006 & 0.19 & 0.19 & 0 & 307.58 & 11 & 1597 \\ Random (60) & 0.17 & 0.17 & 0 & 1829.16 & 12 & 1288 & 0.17 & 0.17 & 0 & 301.42 & 12 & 1331 \\ Random (65) & 0.09 & 0.07 & 0.2 & 3600.03 & 13 & 3537 & 0.09 & 0.09 & 0 & 1384.99 & 13 & 2922 \\ Random (70) & 0.15 & 0.11 & 0.23 & 3600.02 & 14 & 2171 & 0.15 & 0.15 & 0 & 1233.56 & 14 & 3199 \\ Random (80) & 0.06 & 0.01 & 0.89 & 3600.05 & 16 & 2074 & 0.06 & 0.01 & 0.91 & 3600.01 & 16 & 3339 \\ Random (100) & 0.05 & 0 & 0.91 & 3600.02 & 20 & 890 & 0.05 & 0 & 0.91 & 3600.01 & 20 & 1313 \\ \hline \multicolumn{11}{c}{\(M_{1}\) with 30\% controllers density} & \multicolumn{6}{c}{\(M_{2}\) with 30\% controllers density} \\ \cline{2-11}  & Solution & Best bound & **MipCap** & **CPU** & **NC** & _B\& _R_B_ nodes & Solution & Best bound & **MipCap** & **CPU** & **NC** & _B\& _R_B_ nodes \\ \hline Random (20) & 0.19 & 0.19 & 0 & 2.12 & 6 & 1 & 0.19 & 0.19 & 0 & 0.82 & 6 & 1 \\ Random (25) & 0.12 & 0.12 & 0 & 19.56 & 8 & 1436 & 0.12 & 0.12 & 0 & 8.06 & 8 & 4438 \\ Random (30) & 0.13 & 0.13 & 0 & 27.05 & 9 &ranging from 20 nodes to 70 nodes, achieving near-optimal solutions for the 80-node and 100-node networks, respectively. It is also noticeable that model \(M_{2}\) is more efficient for all networks analyzed in this scenario, either in terms of CPU computing times or in the MipGaps obtained for networks where an optimal solution could not be found. In the 40% controller density scenario, it can be seen that model \(M_{1}\) obtains optimal solutions for networks from 20 nodes up to the network with 70 nodes, finding near-optimal solutions for the 80-node and 100-node networks, respectively. In contrast, model \(M_{2}\) achieved optimal solutions for all the studied networks. In this case, model \(M_{2}\) is more efficient in terms of CPU computing times for all studied networks. As it can be observed in Table 4, the model reporting better results is the one for the scenario using 40% controller density, the following study was conducted using \(M_{3}\) in that scenario. This evaluates the behavior of this model, which is similar to \(M_{2}\) but with weights on its components for switch-controller communication and controller-controller communication (note that Model \(M_{2}\) obtained the best results in the previous review). Firstly, it is noted that for \(\alpha=1\), the complexity of the evaluated model is drastically lower because, in this case, the latency of communications between controllers is not considered in the problem under study. Therefore, no further observations or comparisons can be made against this scenario. The first analysis to be conducted is focused on the benchmark networks.