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In this section, the set of 868 LON samplings (for each LON algorithm, totalling 1,736) for QAPLIB is used. Recall that these are associated with 124 QAPLIB instances, each having seven LONs produced per algorithm (differentiated by sampling parameter configuration). Table 5 shows the algorithmic performance prediction model summaries, indicating the strength of the models in terms of R2 and MSE. Tables 6 and 7 record predictor rankings for the random forest models for Markov-Chain LON Samples and Snowball LON Samples, respectively. Rows 1 and 5 of Table 5 (linear regression to predict ILS response) show that neither Markov-Chain nor Snowball LON features build a good model. The R2 values are just too weak. However, when we look at the equivalent random forest models (rows 3 and 7) we have strong models with around 64% (Markov-Chain) and 80% (Snowball) of variance explained, and very small relative MSE values. This could reflect the capability of regression trees to capture nonlinearity and complex interactions between predictors. The same trend is seen in the tabu search prediction models in Table 5. Linear models (rows 2 and 6) are weak, with small R2 and comparably larger MSE. The random forest results (rows 4 and 8) are strong, with 90% or over variance explained by features of LONs produced by both LON construction algorithms.

Table 5:
Linear and random forest regression model summary statistics for full QAPLIB LON set. R2 and MSE are given.
LON MethodRegressionResponse VariableR2MSE
Markov-Chain Linear ILS 0.043 0.002 
Markov-Chain Linear TS 0.180 0.081 
Markov-Chain RandomForest ILS 0.645 0.000 
Markov-Chain RandomForest TS 0.925 0.008 
Snowball Linear ILS 0.057 0.003 
Snowball Linear TS 0.252 0.029 
Snowball RandomForest ILS 0.804 0.000 
Snowball RandomForest TS 0.922 0.003 
LON MethodRegressionResponse VariableR2MSE
Markov-Chain Linear ILS 0.043 0.002 
Markov-Chain Linear TS 0.180 0.081 
Markov-Chain RandomForest ILS 0.645 0.000 
Markov-Chain RandomForest TS 0.925 0.008 
Snowball Linear ILS 0.057 0.003 
Snowball Linear TS 0.252 0.029 
Snowball RandomForest ILS 0.804 0.000 
Snowball RandomForest TS 0.922 0.003 
Table 6:
Predictor rankings for main QAPLIB RF models—Markov-Chain LON samples. Fitness features in italics.
Predicting ILSPredicting TS
mean fitness mean fitness 
funnel-floor fitness funnel-floor fitness 
number optima edges 
incoming global diameter 
edges number optima 
diameter out-degree 
out-degree incoming global 
Predicting ILSPredicting TS
mean fitness mean fitness 
funnel-floor fitness funnel-floor fitness 
number optima edges 
incoming global diameter 
edges number optima 
diameter out-degree 
out-degree incoming global 
Table 7:
Predictor rankings for main QAPLIB RF models using Snowball LONs.
Predicting ILSPredicting TS
mean fitness mean fitness 
maximum HC paths fitness correlation 
fitness correlation maximum HC paths 
mean HC length weight disparity 
weight disparity maximum HC length 
maximum HC length mean HC length 
weight loops weight loops 
Predicting ILSPredicting TS
mean fitness mean fitness 
maximum HC paths fitness correlation 
fitness correlation maximum HC paths 
mean HC length weight disparity 
weight disparity maximum HC length 
maximum HC length mean HC length 
weight loops weight loops 

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