# Edge-Driven Multiple Trajectory Attention Model for Vehicle Routing Problems
## 4 Experiments
### Comparison Results
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline \multirow{2}{*}{Method} & \multirow{2}{*}{Length} & TSP20 & \multicolumn{3}{c}{TSP50} & \multicolumn{3}{c}{TSP100} \\  & & Opt.gap & Time & Length & Opt.gap & Time & Length & Opt.gap & Time \\ \hline LKH & 3.83 & 0.00\% & 42 s & 5.69 & 0.00\% & 7 m & 7.76 & 0.00\% & 26 m \\ Gurobi & 3.83 & 0.00\% & 9 s & 5.69 & 0.00\% & 2 m & 7.76 & 0.00\% & 19 m \\ Concorde & 3.83 & 0.00\% & 5 m & 5.69 & 0.00\% & 13 m & 7.76 & 0.00\% & 1 h \\ \hline Google OR Tools & 3.86 & 0.78\% & 1 m & 5.85 & 2.81\% & 5 m & 8.06 & 3.87\% & 25 m \\ ACO & 3.86 & 0.78\% & 1 m & 5.84 & 2.64\% & 5 m & 8.05 & 3.74\% & 24 m \\ EMA & 3.85 & 0.52\% & 1 m & 5.80 & 1.93\% & 4 m & 7.99 & 2.96\% & 22 m \\ PtrNet & 3.90 & 1.83\% & - & 5.89 & 3.51\% & - & 8.13 & 4.77\% & - \\ GCN & 3.87 & 1.04\% & 11 m & 5.83 & 2.46\% & 19 m & 8.09 & 4.25\% & 42 m \\ DACT & **3.84** & **0.26\%** & 45 s & 5.75 & 1.05\% & 3 m & 7.89 & 1.68\% & 6 m \\ AM & 3.86 & 0.78\% & 2 s & 5.78 & 1.58\% & 4 s & 8.11 & 4.51\% & 10 s \\ POMO & 3.85 & 0.52\% & 2 s & 5.75 & 1.05\% & 4 s & 7.87 & 1.42\% & 11 s \\ E-MTAM & **3.84** & **0.26\%** & 9 s & **5.71** & **0.35\%** & 32 s & **7.82** & **0.77\%** & 1 m \\ \hline \hline \multirow{2}{*}{Method} & \multirow{2}{*}{Length} & CVRP20 & \multicolumn{3}{c}{CVRP50} & \multicolumn{3}{c}{CVRP100} \\  & & Opt.gap & Time & Length & Opt.gap & Time & Length & Opt.gap & Time \\ \hline LKH & 6.10 & 0.00\% & 2 h & 10.38 & 0.00\% & 8 h & 15.65 & 0.00\% & 14 h \\ Gurobi & 6.10 & 0.00\% & 1 m & & - & & - & - & \\ \hline Google OR Tools & 6.45 & 5.74\% & 2 m & 11.31 & 8.96\% & 15 m & 17.21 & 9.97\% & 1 h \\ ACO & 6.41 & 5.08\% & 2 m & 11.22 & 8.09\% & 12 m & 17.05 & 8.95\% & 52 m \\ EMA & 6.39 & 4.75\% & 2 m & 11.13 & 7.23\% & 12 m & 16.88 & 7.86\% & 44 m \\ PtrNet & 6.41 & 5.08\% & - & 11.15 & 7.42\% & - & 16.96 & 8.37\% & - \\ GCN & 6.41 & 5.08\% & 19 m & 11.02 & 6.17\% & 37 m & 16.77 & 7.16\% & 1 h \\ DACT & 6.20 & 1.64\% & 27 s & 10.59 & 2.02\% & 1 m & 16.17 & 3.32\% & 3 m \\ AM & 6.40 & 4.92\% & 2 s & 10.99 & 5.88\% & 5 s & 16.81 & 7.41\% & 13 s \\ POMO & 6.31 & 3.44\% & 2 s & 10.83 & 4.34\% & 6 s & 16.36 & 4.54\% & 15 s \\ E-MTAM & **6.14** & **0.66\%** & 13 s & **10.46** & **0.77\%** & 41 s & **15.81** & **1.02\%** & 2 m \\ \hline \hline \multirow{2}{*}{Method} & \multirow{2}{*}{Prize} & OP20 & \multicolumn{3}{c}{OP50} & \multicolumn{3}{c}{OP100} \\  & & Opt.gap & Time & Prize & Opt.gap & Time & Prize & Opt.gap & Time \\ \hline Gurobi & 5.39 & 0.00\% & 16 m & 16.19 & 0.00\% & 1 h & 33.19 & 0.00\% & 4 h \\ \hline AM & 5.27 & 2.23\% & 2 s & 15.76 & 2.66\% & 4 s & 31.84 & 4.07\% & 10 s \\ POMO & 5.33 & 1.11\% & 2 s & 15.93 & 1.61\% & 4 s & 32.79 & 1.21\% & 12 s \\ E-MTAM & **5.35** & **0.74\%** & 12 s & **16.06** & **0.80\%** & 39 s & **32.85** & **1.02\%** & 1 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison results with exact, heuristic, and DRL-based baselines on TSPs, CVRPs and OPs. converges in the later stages, demonstrating the effectiveness of our training framework. Furthermore, E-MTAM exhibits a faster learning speed, although it does not attain a shorter tour length in the early stages (especially on CVRP20 and CVRP50). E-MTAM ultimately delivers superior results across all problems, approaching the best results attained by LKH (indicated by the horizontal dashed line). Notably, as the problem size increases, the advantages of E-MTAM become more pronounced.