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）是求解多目标优化问题最著名的算法之一。近年来，该算法首次获得了数学运行时间保证，但仅限于合成基准问题。本研究首次为经典优化问题——NP完全的bi-objective最小生成树问题——给出了经过验证的性能保证。具体而言，我们证明，当种群规模 $N \ge 4((n-1) w_{\max} + 1)$ 时，NSGA-II能够在期望的 $O(m^2 n w_{\max} \log(n w_{\max}))$ 次迭代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。这一结果通过数学手段证实了NSGA-II在经验中观察到的良好性能，同时表明对该算法的数学分析不仅适用于合成基准问题，还可扩展至更复杂的组合优化问题。作为附带结果，我们针对bi-objective最小生成树问题，对全局SEMO算法的性能提出了新的分析方法，将先前的最佳结果优化了因子 $|F|$（即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我们的观察：这两种多目标进化算法是并行而非串行地寻找不同极值点，而先前证明中均采用串行假设。