On the Bias and Performance of the Edge-Set Encoding

The edge-set encoding is a direct encoding for trees which directly represents trees as sets of edges. In contrast to indirect representations, where usually standard operators are applied to a list of strings and the resulting phenotype is constructed by an appropriate genotype-phenotype mapping, encoding-specific initialization, crossover, and mutation operators have been developed for the edge-set encoding, which are directly applied to trees. There are two different variants of operators: heuristic versions that consider the weights of the edges and non-heuristic versions. An investigation into the bias of the different variants of the operators shows that the heuristic variants are biased towards the minimum spanning tree (MST), that means solutions similar to the MST are favored. In contrast, non-heuristic versions are unbiased. The performance of edgesets is investigated for the optimal communication spanning tree (OCST) problem. Results are presented for randomly created problems as well as for test instances from the literature. Although optimal solutions for the OCST problem are similar to the MST, evolutionary algorithms using the heuristic crossover operator fail if the optimal solution is only slightly different from the MST. The non-heuristic version shows similar performance as the network random key encoding, which is an unbiased indirect encoding and is used as a benchmark. With proper parameter setting the heuristic version of the mutation operator shows good results for the OCST problem as it can make use of the fact that optimal solutions of the OCST problem are similar to the MST. The results suggest that the heuristic crossover operator of the edge-set encoding should not be used for tree problems as its bias towards the MST is too strong

Making the Edge-Set Encoding Fly by Controlling the Bias of its Crossover Operator

The edge-set encoding is a direct tree encoding which applies search operators directly to trees represented as sets of edges. There are two variants of crossover operators for the edge-set encoding: With heuristics that consider the weights of the edges, or without heuristics. Due to a strong bias of the heuristic crossover operator towards the minimum spanning tree (MST) a population of solutions converges quickly towards the MST and EAs using this operator show low performance when used for tree optimization problems where the optimal solution is not the MST

On the bias and performance of the edge-set encoding

The edge-set encoding of trees directly represents trees as sets of their edges. Nonheuristic operators for edge-sets manipulate trees's edges without regard for their weights, while heuristic operators consider edges' weights when including or excluding them. In the latter case, the operators generally favor edges with lower weights, and they tend to generate trees that resemble minimum spanning trees. This bias is strong, which suggests that evolutionary algorithms (EAs) that employ heuristic operators will succeed when optimum solutions resemble minimum spanning trees (MSTs) but fail otherwise. The one-max tree problem is a scalable test problem for trees where the optimum solution can be predefined. Heuristic operators for edge-sets fail when optimum solutions are random trees or stars. Similarly, for the optimal communication spanning tree (OCST) problem, heuristic operators are efficient only for problem instances where optimal solutions are slightly different from MSTs. In contrast, for both problems the performance of nonheuristic operators is approximately independent of the type of the optimal solution. Therefore, heuristic operators for edge-sets should be used only if optimal solutions closely resemble MSTs. If optimal solutions have low or no bias towards MSTs, heuristic operators for edge-sets fail, and nonheuristic operators should be preferred