This paper revisits the 2-approximation algorithm for $k$-MST presented by Garg in light of a recent paper of Paul et al.. In the $k$-MST problem, the goal is to return a tree spanning $k$ vertices of minimum total edge cost. Paul et al. extend Garg's primal-dual subroutine to improve the approximation ratios for the budgeted prize-collecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the $k$-MST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function.

翻译：本文结合Paul等人近期的工作，重新审视了Garg提出的k-MST问题2-近似算法。k-MST问题的目标是返回一棵包含k个顶点且总边权最小的树。Paul等人扩展了Garg的原始对偶子程序，改进了预算约束下的奖收集旅行商问题和最小生成树问题的近似比。我们借鉴其算法与分析，给出了Garg结果的更简洁版本。此外，我们引入了核这一新概念，使算法各阶段的可视化更简便，剪枝阶段的理解更清晰。其他显著更新包括：给出k-MST问题的线性规划形式（含伪代码），用更简单的“中性集”概念替代Garg的着色方案，以及提供显式的势函数。