We contribute to the efficient approximation of the Pareto-set for the classical $\mathcal{NP}$-hard multi-objective minimum spanning tree problem (moMST) adopting evolutionary computation. More precisely, by building upon preliminary work, we analyse the neighborhood structure of Pareto-optimal spanning trees and design several highly biased sub-graph-based mutation operators founded on the gained insights. In a nutshell, these operators replace (un)connected sub-trees of candidate solutions with locally optimal sub-trees. The latter (biased) step is realized by applying Kruskal's single-objective MST algorithm to a weighted sum scalarization of a sub-graph. We prove runtime complexity results for the introduced operators and investigate the desirable Pareto-beneficial property. This property states that mutants cannot be dominated by their parent. Moreover, we perform an extensive experimental benchmark study to showcase the operator's practical suitability. Our results confirm that the sub-graph based operators beat baseline algorithms from the literature even with severely restricted computational budget in terms of function evaluations on four different classes of complete graphs with different shapes of the Pareto-front.

翻译：我们采用进化计算方法，致力于高效逼近经典$\mathcal{NP}$-难的多目标最小生成树问题（moMST）的帕累托解集。具体而言，基于前期工作，我们分析了帕累托最优生成树的邻域结构，并根据所获得的洞见设计了几种高度偏置的基于子图的变异算子。简言之，这些算子用局部最优子树替换候选解中（不）连通的子树。上述（偏置）步骤通过将克鲁斯卡尔单目标MST算法应用于子图的加权和标量化来实现。我们证明了所引入算子的运行时复杂性结果，并研究了理想的帕累托优势性质——该性质表明变异个体不能被其父代支配。此外，我们进行了广泛的实验基准研究以展示算子的实际适用性。实验结果证实，即使在函数评估预算严重受限的情况下，基于子图的算子在四类不同帕累托前沿形状的完全图上全面超越了文献中的基线算法。