Parameterized Runtime Analyses of Evolutionary Algorithms for the Euclidean Traveling Salesperson Problem

Parameterized runtime analysis seeks to understand the influence of problem structure on algorithmic runtime. In this paper, we contribute to the theoretical understanding of evolutionary algorithms and carry out a parameterized analysis of evolutionary algorithms for the Euclidean traveling salesperson problem (Euclidean TSP). We investigate the structural properties in TSP instances that influence the optimization process of evolutionary algorithms and use this information to bound the runtime of simple evolutionary algorithms. Our analysis studies the runtime in dependence of the number of inner points $k$ and shows that $(\mu + \lambda)$ evolutionary algorithms solve the Euclidean TSP in expected time $O((\mu/\lambda) \cdot n^3\gamma(\epsilon) + n\gamma(\epsilon) + (\mu/\lambda) \cdot n^{4k}(2k-1)!)$ where $\gamma$ is a function of the minimum angle $\epsilon$ between any three points. Finally, our analysis provides insights into designing a mutation operator that improves the upper bound on expected runtime. We show that a mixed mutation strategy that incorporates both 2-opt moves and permutation jumps results in an upper bound of $O((\mu/\lambda) \cdot n^3\gamma(\epsilon) + n\gamma(\epsilon) + (\mu/\lambda) \cdot n^{2k}(k-1)!)$ for the $(\mu+\lambda)$ EA.