## 4 Experimental Comparison
### Experimental Results on Common UDG Instances  
Table 2 presents the experimental outcomes of the PCC\({}^{2}\)LS algorithm alongside other comparative algorithms on common UDG instances. Each column retains the same definitions as in Table 1, where the \"Min\" column denotes the best solution achieved by each algorithm over 10 runs using distinct random seeds.  
From Table 2, it can be observed that the greedy algorithm performs poorly on most of the problems compared to the other algorithms. The ACO and ACO + PCS algorithms obtain the best solution on only two instances (400_80_80) and (400_80_100). At the same time, the GRASP algorithm achieved the best solution in six instances. The VDNS algorithm found the same 18 best solutions within the 300 and 30 s time constraints, while the MSLS algorithm found the 28 best solutions. Although the PCC\({}^{2}\)LS algorithm failed to find the best solution in the (3000_400_210) instance, it achieved the best solution in all other instances. The PCC\({}^{2}\)LS algorithm significantly outperformed the other algorithms in terms of average solution. Therefore, in summary, the PCC\({}^{2}\)LS algorithm is more advantageous in terms of solution quality despite the longer solution time.  
\begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multirow{2}{*}{**LB**} & \multirow{2}{*}{**LPNMR**} & \multirow{2}{*}{**Greedy**} & \multicolumn{3}{c}{**ACO**} & \multicolumn{3}{c}{**ACO + PCS**} & \multicolumn{3}{c}{**GRASP**} & \multicolumn{3}{c}{**VDNS**} & \multicolumn{3}{c}{**MSLS**} & \multicolumn{3}{c}{**PCC\({}^{2}\)LS**} \\ \cline{3-19}  & & & & **Avg** & **St.** & **Tim** & **Avg** & **St.** & **Tim** & **Avg** & **St.** & **Tim** & **Avg** & **St.** & **Tim** & **Avg** & **St.** & **Tim** & **Avg** & **St.** & **Tim** \\ \hline
40 \(\times\) 200 & 5 & 5 & 10 & 5.8 & 0.6 & 3.2 & 5.3 & 0.45 & 4.1 & 5 & 0 & 7.1 & 5 & \(<\)1 & 5 & 0 & 0.00 & 5 & 0 & 0.00 \\
45 \(\times\) 250 & 5 & 5 & 15 & 5.8 & 0.4 & 3.5 & 5.5 & 0.5 & 4.3 & 5 & 0 & 8.7 & 5 & \(<\)1 & 5 & 0 & 0.00 & 5 & 0 & 0.00 \\
50 \(\times\) 250,1 & 6 & 8 & 15 & 8.1 & 0.54 & 4.8 & 8 & 0 & 6.1 & 7 & 0 & 9.8 & 8 & \(<\)1 & 7 & 0 & 0.00 & 7 & 0.01 \\
50 \(\times\) 250,2 & 6 & 7 & 17 & 7.5 & 0.5 & 5.7 & 7.1 & 0.3 & 6.5 & 7 & 0 & 9.9 & 7 & \(<\)1 & 7 & 0 & 0.01 & 7 & 0 & 0.00 \\
55 \(\times\) 250 & 6 & 8 & 20 & 8.8 & 0.98 & 5.6 & 8.3 & 0.45 & 7.3 & 8 & 0 & 9.8 & 8 & \(<\)1 & 8 & 0 & 0.01 & 8 & 0.01 \\
60 \(\times\) 400 & 5 & 7 & 15 & 7 & 0 & 6.1 & 7 & 0 & 9.1 & 7 & 0 & 12.7 & 7 & \(<\)1 & 7 & 0 & 0.01 & 7 & 0 & 0.00 \\
70 \(\times\) 250 & 11 & 13 & 32 & 14.2 & 0.74 & 11 & 13.9 & 1.04 & 13.5 & 12 & 0 & 11.5 & 13 & \(<\)1 & 12 & 0 & 0.00 & **12** & 0.00 \\
80 \(\times\) 500 & 7 & 9 & 20 & 10 & 0.44 & 12 & 9.8 & 0.4 & 16.9 & **9** & 0 & 15.1 & **9** & 0 & 0.00 & 9 & 0 & 0.01 \\
90 \(\times\) 600 & 7 & 10 & 19 & 10.9 & 0.83 & 14 & 10.6 & 1.01 & 17.3 & 9 & 0 & 17.2 & 10 & \(<\)1 & 9 & 0 & 0.01 & 9 & 0 & 0.01 \\ average & 6.4 & 8.0 & 18.1 & 8.7 & & & 8.3 & & & 7.7 & & 8.0 & & 7.7 & & 7.7 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Results of each algorithm on LPNMR’09 instances.