An evolutionary approach for tuning parametric Esau and Williams heuristics

Abstract

Owing to its inherent difficulty, many heuristic solution methods have been proposed for the capacitated minimum spanning tree problem. On the basis of recent developments, it is clear that the best metaheuristic implementations outperform classical heuristics. Unfortunately, they require long computing times and may not be very easy to implement, which explains the popularity of the Esau and Williams heuristic in practice, and the motivation behind its enhancements. Some of these enhancements involve parameters and their accuracy becomes nearly competitive with the best metaheuristics when they are tuned properly, which is usually done using a grid search within given search intervals for the parameters. In this work, we propose a genetic algorithm parameter setting procedure. Computational results show that the new method is even more accurate than an enumerative approach, and much more efficient.

Appendices

Appendix A

Analysis set choice

The strategy used to select the analysis set can affect the performance of a GA. We test four analysis set selection strategies for EWG, EWLS, and EWR.

The first analysis set is obtained through the general and deterministic (G/D) strategy. It is general, because it includes instances from all test problem sets. The first two instances with the two smallest capacities are deterministically selected from the sets cm50, cm100, and cm200, which totally makes six instances. We qualify these three sets as more difficult than the problem sets tc40, tc80, te40, and te80, since the EW and its parametric enhancement end up with larger deviations on them.

The remaining three strategies are dedicated in the sense that they determine the analysis instances particularly from one of the three problem groups tc, te, and cm. We believe that this can give the genetic search procedure a higher chance to learn better the set's structural characteristics, but at the expenses of three executions of the GA. Moreover, we use three rules to determine the dedicated instances in the analysis set. The first rule is deterministic. We choose the first two instances from each one of the two smallest capacity groups, as in G/D; this is what we call D/D. The second rule selects the most difficult instances and it is denoted by D/M. The two instances for which EW3 produces the smallest improvements over EW are selected from the test problem groups. Finally, the third rule is based on the random selection of two instances from each group, and it is referred as D/R. Notice that the size of the analysis sets are four for tc and te, and six for the cm instances, for any analysis set strategy. The five parameter vectors forming the set are listed in Table A1, for the four analysis set selection strategies.

Table A1 Results of the genetic search: parameters versus analysis sets

The average percentage deviations and CPU times computed running EWBF3 and EWLS are summarized in Table A2. Note that they are obtained according to four distinct analysis set strategy for 0, 1, 2, 3, 4, and 5 search cubes. The first column denotes the sets of test problem sets. The second column includes the four analysis set selection strategies. Notice that the average relative percent improvements made on EW and total CPU time in seconds are reported; they are computed with the new re-implementations.

Table A2 The effect of the analysis set and the number of search cubes on the performance of EWLS

The results show that a dedicated analysis set can produce slightly better results than the general analysis set. In some cases, the dedicated algorithm produces even better results than EWBF3; these are marked in bold in Table A2. Moreover, the improvements seem to be larger when the instances solved are large (namely, cm200). The most effective analysis set choice seems to be the D/M, but the results obtained with other strategies are not substantially different.

Moreover, results obtained by increasing the number of cubes considered in the LS step are quite interesting. In fact, the first two cubes introduce major improvements in the solution quality, whereas a higher number of cubes result in marginal improvements for a relatively higher computational cost. This behaviour shows that EWG is able to detect promising areas in parameter space fairly well. Whenever we move unnecessarily far away from these regions, by considering a higher number of cubes, the performance of the method does not improve anymore.

Appendix B

Initialization of the randomized prohibition heuristic

The RP heuristic is effective even by considering a very limited number of EW3 iterations, as reported in Section 4. However, we think we should test the performance of the standalone method and we should verify if the solution quality is dependent or not on the saving list quality.

The first experiment consists in initializing RP with a random saving list. Three random parameters are generated in [0, 2] and the corresponding EW3 saving list initializes 195 iterations of RP. The solution quality obtained is definitely not satisfactory, in fact a single run of EW is on the average better than 195 iterations of the randomly initialized RP algorithm.

On the other hand, if the EW3 saving list is initialized with the EW parameters (ie, α=1, β=γ=0), the RP's average improvement is 3.86% with respect to a single EW execution. If the saving list is computed by considering higher quality parameters, such as the three best parameter vectors obtained after the genetic algorithm (ie, the first three parameter sets in ) and each saving list initializes 195/3=65 RP iterations, the average improvement increases to 4.16%. RP proves to be strongly dependent on the quality of the saving list considered. When the saving list is initialized by even better parameters, as the best parameter set found by EWLS, even with 65 RP iterations are performed, the average improvement is 5.11%. If the parameter vectors are the best three found and 65 RP iterations are run for each corresponding saving list, the average improvement increases to 5.50%.

For the sake of completeness, we pushed this latter configuration by considering a larger number of iterations. We executed 1500 and 3000 iterations of the RP algorithm. The average percentage improvement grows to 5.99% for 3000 iterations, but the improvements beyond 1500 iterations are considerably smaller.