We present improved learning-augmented algorithms for finding an approximate minimum spanning tree (MST) for points in an arbitrary metric space. Our work follows a recent framework called metric forest completion (MFC), where the learned input is a forest that must be given additional edges to form a full spanning tree. Veldt et al. (2025) showed that optimally completing the forest takes $Ω(n^2)$ time, but designed a 2.62-approximation for MFC with subquadratic complexity. The same method is a $(2γ+ 1)$-approximation for the original MST problem, where $γ\geq 1$ is a quality parameter for the initial forest. We introduce a generalized method that interpolates between this prior algorithm and an optimal $Ω(n^2)$-time MFC algorithm. Our approach considers only edges incident to a growing number of strategically chosen ``representative'' points. One corollary of our analysis is to improve the approximation factor of the previous algorithm from 2.62 for MFC and $(2γ+1)$ for metric MST to 2 and $2γ$ respectively. We prove this is tight for worst-case instances, but we still obtain better instance-specific approximations using our generalized method. We complement our theoretical results with a thorough experimental evaluation.

翻译：本文针对任意度量空间中点的近似最小生成树（MST）求解问题，提出了改进的学习增强算法。我们的工作基于近期提出的度量森林补全（MFC）框架，其中学习得到的输入是一个森林，需要通过添加边来构成完整的生成树。Veldt等人（2025）证明了最优补全森林需要$Ω(n^2)$时间，但设计了一个具有次二次复杂度的2.62近似MFC算法。该方法对原始MST问题可达到$(2γ+1)$近似比，其中$γ≥1$是初始森林的质量参数。我们提出了一种广义方法，能够在现有算法与最优的$Ω(n^2)$时间MFC算法之间进行插值。该方法仅考虑与逐步增加的策略性选取“代表点”相关联的边。我们分析的一个推论是：将先前算法对MFC的近似比从2.62提升至2，对度量MST的近似比从$(2γ+1)$提升至$2γ$。我们证明该结果在最坏情况实例下是紧的，但通过广义方法仍可获得更优的实例特定近似比。最后，我们通过系统的实验评估对理论结果进行了补充。