## 5 Experimental Results
### Comparison of NewKLS_FVS with Other Algorithms
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Instance** & **Best** & **Worst** & **Average** & **Instance** & **Best** & **Worst** & **Average** \\ \hline auna.col & 113 & 100 & 104 & miles1000.col & 14 & 12 & 13 \\ david.col & 54 & 49 & 51 & miles1500.col & 9 & 8 & 8 \\ DSJC125.1.col & 40 & 37 & 37 & mulsol.i.1.col & 38 & 29 & 32 \\ DSJC125.5.col & 10 & 9 & 9 & mulsol.i.2.col & 45 & 39 & 42 \\ DSJC125.9.col & 5 & 5 & 5 & mulsol.i.3.col & 47 & 39 & 42 \\ fpsol2.i.1.col & 90 & 66 & 73 & mulsol.i.4.col & 47 & 39 & 42 \\ fpsol2.i.2.col & 133 & 116 & 123 & mulsol.i.5.col & 48 & 38 & 42 \\ fpsol2.i.3.col & 133 & 109 & 116 & myciel3.col & 8 & 8 & 8 \\ games120.col & 36 & 34 & 35 & myciel4.col & 16 & 15 & 15 \\ homer.col & 331 & 325 & 327 & myciel6.col & 40 & 34 & 37 \\ huck.col & 45 & 41 & 42 & myciel7.col & 54 & 46 & 50 \\ inithx.i.1.col & 131 & 124 & 128 & queen5\_5.col & 7 & 7 & 7 \\ inithx.i.2.col & 148 & 138 & 143 & queen6\_6.col & 9 & 9 & 9 \\ inithx.i.3.col & 145 & 134 & 139 & queen7\_7.col & 10 & 10 & 10 \\ jean.col & 56 & 52 & 53 & queen8\_8.col & 12 & 11 & 11 \\ latin\_square\_10.col & 7 & 6 & 6 & queen8\_12.col & 14 & 13 & 13 \\ le450\_5a.col & 61 & 57 & 58 & queen9\_9.col & 13 & 12 & 12 \\ le450\_5b.col & 60 & 57 & 58 & queen10\_10.col & 14 & 13 & 13 \\ le450\_5c.col & 40 & 36 & 37 & queen11\_11.col & 15 & 14 & 14 \\ le450\_5d.col & 41 & 37 & 38 & queen12\_12.col & 17 & 15 & 15 \\ le450\_15b.col & 54 & 48 & 50 & queen13\_13.col & 18 & 16 & 16 \\ le450\_15c.col & 28 & 26 & 27 & queen14\_14.col & 18 & 17 & 17 \\ le450\_15d.col & 28 & 26 & 26 & queen15\_15.col & 19 & 18 & 18 \\ le450\_25a.col & 61 & 53 & 56 & queen16\_16.col & 21 & 19 & 19 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for simulated annealing (SA) and a CPU-time limit of 30 min for 15 runs. In order to show the performance and efficiency of various algorithms, experiments are conducted to test the algorithms (NewkLS_FVS, k-opt-LS, VNS1, VNS2, SA) on graphs huck.col, inithx.i.1.col, le450_25a.col, zeroin.i.1.col and school1_nsh.col. The convergence curves of these algorithms are reported on both running times (with a CPU-time limit of 10 min; see Figure 1) and iteration numbers (within 100 iterations; see Figure 2). The experimental results show that NewkLS_FVS outperforms other algorithms on graphs huck.col, le450_25a.col, zeroin.i.1.col and school1_nsh.col almost at any moment. The solution quality is better than any other compared algorithm after 3 min on inithx.i.1.col. Additionally, the solution quality of NewkLS_FVS is better than those of any other compared algorithm on all tested graphs after 20 iterations. It can be seen that the performance of the algorithm, k-opt-LS, is better than VNS1, VNS2 and SA on almost all these instances, and the convergence performance of k-opt-LS is improved by using the proposed approaches. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Instance** & **Best** & **Worst** & **Average** & **Instance** & **Best** & **Worst** & **Average** \\ \hline le450\_25b.col & 56 & 52 & 53 & school1.col & 26 & 23 & 23 \\ le450\_25c.col & 30 & 26 & 28 & school1\_nsh.col & 25 & 23 & 23 \\ le450\_25d.col & 30 & 27 & 27 & zeroin.i.1.col & 51 & 37 & 42 \\ miles250.col & 61 & 56 & 58 & zeroin.i.2.col & 67 & 54 & 59 \\ miles500.col & 31 & 28 & 29 & zeroin.i.3.col & 63 & 53 & 57 \\ miles750.col & 19 & 18 & 18 & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: _Cont._  
Figure 1: Comparison of the algorithms with a CPU-time limit of 10 min. (**a**) inithx.i.1.col; (**b**) le450_25a.col; (**c**) school1_nsh.col; (**d**) zeroin.i.1.col.