The $(1+(λ,λ))$ Genetic Algorithm for Permutations

The $(1+(\lambda,\lambda))$ genetic algorithm is a bright example of an evolutionary algorithm which was developed based on the insights from theoretical findings. This algorithm uses crossover, and it was shown to asymptotically outperform all mutation-based evolutionary algorithms even on simple problems like OneMax. Subsequently it was studied on a number of other problems, but all of these were pseudo-Boolean. We aim at improving this situation by proposing an adaptation of the $(1+(\lambda,\lambda))$ genetic algorithm to permutation-based problems. Such an adaptation is required, because permutations are noticeably different from bit strings in some key aspects, such as the number of possible mutations and their mutual dependence. We also present the first runtime analysis of this algorithm on a permutation-based problem called Ham whose properties resemble those of OneMax. On this problem, where the simple mutation-based algorithms have the running time of $\Theta(n^2 \log n)$ for problem size $n$, the $(1+(\lambda,\lambda))$ genetic algorithm finds the optimum in $O(n^2)$ fitness queries. We augment this analysis with experiments, which show that this algorithm is also fast in practice.

The 1/5-th Rule with Rollbacks: On Self-Adjustment of the Population Size in the $)$ GA

Self-adjustment of parameters can significantly improve the performance of evolutionary algorithms. A notable example is the $(1+(\lambda,\lambda))$ genetic algorithm, where the adaptation of the population size helps to achieve the linear runtime on the OneMax problem. However, on problems which interfere with the assumptions behind the self-adjustment procedure, its usage can lead to performance degradation compared to static parameter choices. In particular, the one fifth rule, which guides the adaptation in the example above, is able to raise the population size too fast on problems which are too far away from the perfect fitness-distance correlation. We propose a modification of the one fifth rule in order to have less negative impact on the performance in scenarios when the original rule reduces the performance. Our modification, while still having a good performance on OneMax, both theoretically and in practice, also shows better results on linear functions with random weights and on random satisfiable MAX-SAT instances.