We study the multi-objective minimum weight base problem, an abstraction of classical NP-hard combinatorial problems such as the multi-objective minimum spanning tree problem. We prove some important properties of the convex hull of the non-dominated front, such as its approximation quality and an upper bound on the number of extreme points. Using these properties, we give the first run-time analysis of the MOEA/D algorithm for this problem, an evolutionary algorithm that effectively optimizes by decomposing the objectives into single-objective components. We show that the MOEA/D, given an appropriate decomposition setting, finds all extreme points within expected fixed-parameter polynomial time in the oracle model, the parameter being the number of objectives. Experiments are conducted on random bi-objective minimum spanning tree instances, and the results agree with our theoretical findings. Furthermore, compared with a previously studied evolutionary algorithm for the problem GSEMO, MOEA/D finds all extreme points much faster across all instances.

翻译：我们研究了多目标最小权重基问题，这是对经典NP难组合问题（如多目标最小生成树问题）的抽象。我们证明了非支配前沿凸包的一些重要性质，例如其近似质量以及极端点数量的上界。利用这些性质，我们首次给出MOEA/D算法在此问题上的运行时分析——该算法通过将目标分解为单目标分量进行有效优化。我们证明，在适当的分解设置下，MOEA/D能在预言机模型中以期望的固定参数多项式时间找到所有极端点，其中参数为目标数量。我们在随机双目标最小生成树实例上进行了实验，结果与理论发现一致。此外，与先前研究的该问题进化算法GSEMO相比，MOEA/D在所有实例上均能更快找到所有极端点。