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It is important to be able to identify instance features which can inform specific algorithm selection. To this end, a decision tree model was trained using the Pihera and TSPMETA feature sets described earlier. A decision tree is a simple but useful technique for creating “rules of thumb ” that can be applied to identify instance traits that may hamper the performance of specific solvers. This process identified key features (and their associated values) which can be used to select the most effective instance-specific algorithms. From the head of the decision tree shown in Fig. 5, two key features were extracted which can be used to identify a significant variance in performance between the EAX and SET-LKH solvers: nn5.sc.max.n, from the Pihera feature set; and mst_dists_median, from the TSPMETA feature set. It can be seen that 72% of instances were solved significantly faster by EAX than either of the SET-LKH heuristics and, in a further 16%, no significan difference was found (cf. Table 5 & Fig. 5(a)). The intuitive choice, therefore would be to use EAX by default; however, note that 12% of all instances testec were best solved by the SET-LKH heuristics. It was found that 67% of all in stances exhibited nn5.sc.max.n < 0.71; of these, 93% were solved best by EAX (cf. Figures 5(b) and 6(a)). When nn5.sc.max.n >= 0.71, 71% were solved by SET-LKH heuristics in comparable (29%) or less (42%) time, Fig. 5(c).

Figure 5 It is important to be able to identify instance features which can inform specific algorithm selection. To this end, a decision tree model was trained using the Pihera and TSPMETA feature sets described earlier. A decision tree is a simple but useful technique for creating “rules of thumb ” that can be applied to identify instance traits that may hamper the performance of specific solvers. This process identified key features (and their associated values) which can be used to select the most effective instance-specific algorithms. From the head of the decision tree shown in Fig. 5, two key features were extracted which can be used to identify a significant variance in performance between the EAX and SET-LKH solvers: nn5.sc.max.n, from the Pihera feature set; and mst_dists_median, from the TSPMETA feature set. It can be seen that 72% of instances were solved significantly faster by EAX than either of the SET-LKH heuristics and, in a further 16%, no significan difference was found (cf. Table 5 & Fig. 5(a)). The intuitive choice, therefore would be to use EAX by default; however, note that 12% of all instances testec were best solved by the SET-LKH heuristics. It was found that 67% of all in stances exhibited nn5.sc.max.n < 0.71; of these, 93% were solved best by EAX (cf. Figures 5(b) and 6(a)). When nn5.sc.max.n >= 0.71, 71% were solved by SET-LKH heuristics in comparable (29%) or less (42%) time, Fig. 5(c).

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Fig. 1: Instance type problem domains. Plots (a)—(c) are examples of the RUE, MORPH & NETGEN instances, all with n = 500. The MORPH example, (b), is constructed from (a) and (c), with a = 0.5. Plot (d) shows rat575, a rattled grid of 575 locations; plot (e) is the World instance with 734 town/cities in Uruguay (uy734); and (f) is the VLSI problem denoted as dkg813
Fig. 1: Instance type problem domains. Plots (a)—(c) are examples of the RUE, MORPH & NETGEN instances, all with n = 500. The MORPH example, (b), is constructed from (a) and (c), with a = 0.5. Plot (d) shows rat575, a rattled grid of 575 locations; plot (e) is the World instance with 734 town/cities in Uruguay (uy734); and (f) is the VLSI problem denoted as dkg813
Fig.2: EAX v LKH-+IPT runtime performance results. All plot axes are log- scaled; scatter points are median values of 30 runs of pairwise instances in the form (median(EAX), median(LKH+IPT)); diagonal lines represent equality of median performance; boxplots on axes show distribution of median runtime val- ues Firstly, the relative performances of EAX versus 107.7s. Similar values are returned for the MORP runtime = 7.6s and median LKH+IPT = 116.4s. LKH-+IPT are considered, and results are shown in Fig. 2 and Table 1. It can be seen from Fig. 2(a) and (b) that EAX is vastly superior in solving NETGEN and MORPH instance types to optimality when compared with LKH+IPT. The median value of the boxplot or EAX in Fig. 2(a) (which is the overall median value of all EAX runs for ETGEN) is 7.2 seconds (s), whereas the corresponding value for LKH+IPT is H instances, with median EAX
Fig.2: EAX v LKH-+IPT runtime performance results. All plot axes are log- scaled; scatter points are median values of 30 runs of pairwise instances in the form (median(EAX), median(LKH+IPT)); diagonal lines represent equality of median performance; boxplots on axes show distribution of median runtime val- ues Firstly, the relative performances of EAX versus 107.7s. Similar values are returned for the MORP runtime = 7.6s and median LKH+IPT = 116.4s. LKH-+IPT are considered, and results are shown in Fig. 2 and Table 1. It can be seen from Fig. 2(a) and (b) that EAX is vastly superior in solving NETGEN and MORPH instance types to optimality when compared with LKH+IPT. The median value of the boxplot or EAX in Fig. 2(a) (which is the overall median value of all EAX runs for ETGEN) is 7.2 seconds (s), whereas the corresponding value for LKH+IPT is H instances, with median EAX
For the LIB instances, EAX still performed better than LKH+IPT; how- ever, EAX did not overwhelmingly outperform LKH+IPT as it had for the NETGEN and MORPH instance types. The 3 World instances included in out study (uy734, zi929 and mu1979) were solved significantly better by EAX than LKH-+IPT. Focusing on VLSI only, the median runtime value for EAX was 12.8s, with LKH+IPT returning a median runtime of 59.2s. When considering these with the results in Table 1, it is apparent that EAX solves VLSI instances tc optimality with noticeably greater speed than LKH+IPT.
For the LIB instances, EAX still performed better than LKH+IPT; how- ever, EAX did not overwhelmingly outperform LKH+IPT as it had for the NETGEN and MORPH instance types. The 3 World instances included in out study (uy734, zi929 and mu1979) were solved significantly better by EAX than LKH-+IPT. Focusing on VLSI only, the median runtime value for EAX was 12.8s, with LKH+IPT returning a median runtime of 59.2s. When considering these with the results in Table 1, it is apparent that EAX solves VLSI instances tc optimality with noticeably greater speed than LKH+IPT.
Fig. 3: LKH+GPX2 v EAX runtime performance results. All plot axes are log-scaled; scatter points are median values of 30 runs of pairwise instances (LKH+GPX2, EAX); diagonal lines represent equality of median performance; boxplots on axes show distribution of median runtime values
Fig. 3: LKH+GPX2 v EAX runtime performance results. All plot axes are log-scaled; scatter points are median values of 30 runs of pairwise instances (LKH+GPX2, EAX); diagonal lines represent equality of median performance; boxplots on axes show distribution of median runtime values
Fig. 4: LKH+IPT v LKH+GPX2 runtime performance scatter plots As inferred by their relative performances against EAX, the runtime results for LKH+IPT versus LKH+GPX2 exhibit very little variation. Both Fig. 4 and Table 3 show that, for instances of these sizes and types, and with a stopping criterion of one hour, there exists little appreciable difference in their relative performances. The four plots in Fig. 4 show that almost all scatter points lie very closely to the line of median equality. When this did not hold, for example in Fig. 4(d) where a single point is located at (1329, PAR10) (corresponding to VLSI instance bck2217), the Mann-Whitney test does not always allow rejection of its null hypothesis and so the instance is labelled as ANY.
Fig. 4: LKH+IPT v LKH+GPX2 runtime performance scatter plots As inferred by their relative performances against EAX, the runtime results for LKH+IPT versus LKH+GPX2 exhibit very little variation. Both Fig. 4 and Table 3 show that, for instances of these sizes and types, and with a stopping criterion of one hour, there exists little appreciable difference in their relative performances. The four plots in Fig. 4 show that almost all scatter points lie very closely to the line of median equality. When this did not hold, for example in Fig. 4(d) where a single point is located at (1329, PAR10) (corresponding to VLSI instance bck2217), the Mann-Whitney test does not always allow rejection of its null hypothesis and so the instance is labelled as ANY.
Excluding the RUE instances, the proportion best solved by the classifica- tions EAX or ANY rises to 96.6%. Conversely, considering only the RUE in- stances, we see that the SET-LKH heuristics perform effectively 72.5% of the time when combined with the ANY classification. The VLSI instances, often con- sidered the most intractable or difficult to solve to optimality, are best solved by EAX 68.4%; only 14.0% are solved significantly quicker by SET-LKH solvers.
Excluding the RUE instances, the proportion best solved by the classifica- tions EAX or ANY rises to 96.6%. Conversely, considering only the RUE in- stances, we see that the SET-LKH heuristics perform effectively 72.5% of the time when combined with the ANY classification. The VLSI instances, often con- sidered the most intractable or difficult to solve to optimality, are best solved by EAX 68.4%; only 14.0% are solved significantly quicker by SET-LKH solvers.
Fig. 6: Plots show kernels of ANY, EAX, SET-LKH classed instances for features nn5.sc.max.n (a) and mst_dists_median (b). Vertical lines indicate the threshold values obtained from the decision tree in Fig. 5
Fig. 6: Plots show kernels of ANY, EAX, SET-LKH classed instances for features nn5.sc.max.n (a) and mst_dists_median (b). Vertical lines indicate the threshold values obtained from the decision tree in Fig. 5
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