AI 中文总结
本文针对树上的分数顶点覆盖问题，在边到达的在线模型中提出11/6≈1.83竞争比的算法，在更新预先已知的增量模型中提出1.5竞争比的算法并给出匹配下界。
AI 中文摘要
本文在两个相关模型中研究树上的分数顶点覆盖问题：在线模型与增量模型。在线模型中，树的顶点是预先已知的，边则逐个到达，目标是维护树的一个分数顶点覆盖，即给顶点分配[0,1]区间内的分数权重，使得每条边的两个端点的权重之和至少为1；在每条边到达后，需要修改分数顶点覆盖以覆盖新边，且只能增加分配给顶点的数值。Wang和Wong此前在顶点到达模型中研究过该问题，将其作为滑雪租赁问题的推广，还因其与对偶在线匹配问题的紧密联系而受关注，他们在顶点到达模型中针对一般图给出了1.901竞争比的算法。我们在更通用的边到达模型中针对树给出了竞争比为11/6≈1.83的算法。此外，我们在增量模型中研究分数顶点覆盖问题，该模型中算法预先已知树的所有更新，每次更新后仍需寻找分数顶点覆盖，我们在该模型中给出了竞争比为1.5的算法，并提供了匹配的下界。
英文摘要
In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.
Journal ref34th Annual European Symposium on Algorithms (ESA 2026), Leibniz International Proceedings in Informatics (LIPIcs) 388, pp. 158:1-158:15, 2026
DOI：10.4230/LIPIcs.ESA.2026.158
