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等人近期的一篇论文，重新审视了Garg提出的k-MST问题的2-近似算法。在k-MST问题中，目标是返回一棵总边成本最小的、覆盖k个顶点的树。Paul等人对Garg的原始-对偶子例程进行了扩展，从而改进了带预算的奖赏收集旅行商和最小生成树问题的近似比。我们沿用了他们的算法与分析，给出了Garg结果的一个更清晰的版本。此外，我们引入了“核”（kernel）这一新概念，使得算法的各阶段更易于可视化，且对剪枝阶段的理解更为清晰。其他值得注意的更新包括：给出k-MST问题的线性规划形式、提供伪代码、用更简洁的中性集概念替代Garg使用的着色方案，以及给出一个显式势函数。