The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most prominent algorithms to solve multi-objective optimization problems. Recently, the first mathematical runtime guarantees have been obtained for this algorithm, however only for synthetic benchmark problems. In this work, we give the first proven performance guarantees for a classic optimization problem, the NP-complete bi-objective minimum spanning tree problem. More specifically, we show that the NSGA-II with population size $N \ge 4((n-1) w_{\max} + 1)$ computes all extremal points of the Pareto front in an expected number of $O(m^2 n w_{\max} \log(n w_{\max}))$ iterations, where $n$ is the number of vertices, $m$ the number of edges, and $w_{\max}$ is the maximum edge weight in the problem instance. This result confirms, via mathematical means, the good performance of the NSGA-II observed empirically. It also shows that mathematical analyses of this algorithm are not only possible for synthetic benchmark problems, but also for more complex combinatorial optimization problems. As a side result, we also obtain a new analysis of the performance of the global SEMO algorithm on the bi-objective minimum spanning tree problem, which improves the previous best result by a factor of $|F|$, the number of extremal points of the Pareto front, a set that can be as large as $n w_{\max}$. The main reason for this improvement is our observation that both multi-objective evolutionary algorithms find the different extremal points in parallel rather than sequentially, as assumed in the previous proofs.

翻译：非支配排序遗传算法II（NSGA-II）是解决多目标优化问题最著名的算法之一。近期，该算法首次获得了数学运行时保证，然而这些结果仅针对合成基准问题。本研究首次证明了该算法在经典优化问题——NP完全的二元目标最小生成树问题上的性能保证。具体而言，我们证明当种群规模$N \ge 4((n-1) w_{\max} + 1)$时，NSGA-II能在期望迭代次数$O(m^2 n w_{\max} \log(n w_{\max}))$内计算出帕累托前沿的所有极值点，其中$n$为顶点数，$m$为边数，$w_{\max}$为问题实例中的最大边权。该结果通过数学手段验证了NSGA-II在实验观察中表现出的优良性能，同时表明该算法的数学分析不仅适用于合成基准问题，也能拓展至更复杂的组合优化问题。作为副产品，我们获得了全局SEMO算法在二元目标最小生成树问题上的性能新分析，该结果较先前最优结果提升了因子$|F|$——帕累托前沿的极值点数（该集合规模可达$n w_{\max}$）。改进的主要原因在于：不同于先前证明中假设的串行方式，我们观察到两种多目标进化算法实际上是并行发现不同极值点的。