AI 中文总结
该研究结合LearnOrCover框架与Augmented Greedy算法，提出节点加权施泰纳树和森林的O(log n)近似算法，可用于在线场景，达到最优近似界。
AI 中文摘要
我们针对节点加权施泰纳树和节点加权施泰纳森林提出了一种新的O(log n)近似算法。该算法达到了Klein与Ravi[J. Algorithms '95]给出的、除非P=NP否则无法进一步改进的最优界，且具备在终端对以随机顺序呈现的在线场景中运行的优势。为获得该结果，我们将Gupta、Kehne与Levin[FOCS '21]提出的LearnOrCover框架，以及Berman与Coulston[STOC '97]针对在线边加权施泰纳森林的Augmented Greedy算法相结合；两种算法单独使用均无法满足需求，但它们的分析能相互配合，从而保证我们的算法性能。离线运行时，该算法简化为一种极为简单的随机舍入方案，其核心思想是将节点加权施泰纳森林转化为边加权施泰纳森林，我们希望这一思路能得到更多应用。
英文摘要
We give a new $O(\log n)$ approximation algorithm for Node Weighted Steiner Tree and Node Weighted Steiner Forest. Our algorithm matches the bounds of Klein & Ravi [J. Algorithms '95] which are best possible unless P = NP, but have the advantage that they work in the online setting when the terminal pairs are revealed in random order. To obtain our results, we combine the LearnOrCover framework due to Gupta, Kehne, Levin [FOCS '21] with the Augmented Greedy algorithm of Berman & Coulston [STOC '97] for online edge-weighted Steiner Forest. Neither algorithm suffices on its own, but the analyses dovetail to imply our guarantee. Run offline, the algorithm reduces to a very simple randomized rounding scheme that (in spirit) reduces Node Weighted Steiner Forest to Edge Weighted Steiner Forest, and we hope this idea finds further applications.
