Correction to: Evolutionary Computation in Combinatorial Optimization

Correction to: A. Liefooghe and L. Paquete (Eds.): Evolutionary Computation in Combinatorial Optimization, LNCS 11452, https://doi.org/10.1007/978-3-030-16711-0

1. Chapter 6.

In the originally published version the indexes of some variables in Section 4.1, including Constraints (4) and (5) of the model, include a wrong offset of one position. Some errors occurred in notations of variable indexes in Constraints (4) and (5) conditions of the model in Section 4.1, together with some ambiguities that may lead to misunderstanding for the reader. This was corrected in the updated version.

2. Chapter 11.

The originally published version of the paper “Clarifying the Difference in Local Optima Network Sampling Algorithms” contained an error. The additional text correcting the error has been added below.

Summary

During a re-analysis of the data-set, the lead author realised that she had made analytical errors while computing the results for this paper. This erratum presents the corrected numeric results in Section 3. These correspond to Tables 3 and 4 and Figures 2 and 3 — all of which are from Section 4.2 in the original paper. This report discusses the affect on the main conclusions of the work in the next Section. We found that while the numeric results are changed, most of the conclusions are still generally correct.

Affect on Conclusions

Conclusion 1. We found that the two sampling methods exhibited some agreement in the networks they produced and that we could reject the null hypothesis that they produce completely independent samples. They differed from a descriptive perspective in that walkSample was tuneable and predictable, while optSample varied widely but seemed good at finding hub-and-spoke structure in the local optima space. This conclusion is still correct.

Conclusion 2. The correlations were stronger and clear when considering the features of the LONs obtained using optSample than walkSample. This conclusion is still correct.

Conclusion 3. We also worked on explaining heuristic algorithm performance on the problems using linear and random forest models, and found that the sampled LON features (for both optSample and walkSample) better fit the ILS response variable than the TS one. This conclusion is now reversed, i.e., the sampled LON features better fit the TS response variable than the ILS one.

Conclusion 4. We saw that generally, including both the funnel metric set and the network set would be advantageous in explaining search discrepancies for these two heuristics. This conclusion is still correct.

Conclusion 5. For both optSample and walkSample, the extracted funnel metrics proved useful. This conclusion is still correct with a nuance, i.e. the extracted funnel metrics proved useful with respect to TS as a response variable.

Conclusion 6. Going off the random forest models alone, optSample uniformly had more predictive power than its competitor, for these choices of instances and heuristics. This conclusion is still correct overall; optSample generally had more predictive power, but not uniformly.

Conclusion 7. From the random forest rankings, the most important predictors were those pertaining to fitness in the sampled networks: the fitness of funnel bottoms, and of nodes in general in the network. This hints that perhaps fitness levels in the local optima space are more pertinent to heuristic search than the subset of transition edges sampled by the LON algorithms. This conclusion is still correct.

This work is supported by the UK’s Engineering and Physical Sciences Research Council (grant number EP/J017515/1). Data generated during this research are available from the Stirling Online Repository for Research Data (http://hdl.handle.net/11667/128).

Corrected Results

Table 1. Corrected results for Table 3 in the original paper. $R^2$ values for linear and random forest models to explain heuristic performance variation on the combinatorial problems.

Table 2. Corrected results for Table 4 in the original paper. Predictor rankings for the random forest models.

Fig. 1.

Corrected results for Figure 2 in the original paper. Correlation matrix of performance metrics and optSample-produced LON features. Lower triangle: pairwise scatter plots. Diagonal: density plots. Upper triangle: pairwise Spearman’s rank correlation, $^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$.

Fig. 2.

Corrected results for Figure 3 in the original paper. Correlation matrix of performance metrics and walkSample-produced LON features. Lower triangle: pairwise scatter plots. Diagonal: density plots. Upper triangle: pairwise Spearman’s rank correlation, $^{***}p<0.001$, $^{**}p<0.01$, $^*p<0.05$.