AI 中文总结
研究全在线模型下一般图的分数匹配问题，扩展经典注水算法，证明其竞争比为$2-\sqrt{2}$，融入新思想后竞争比提升至$0.599$，还改进了分数全在线匹配的难度上界。
AI 中文摘要
本文研究了Huang等人（JACM 2020）全在线模型下一般图上的分数匹配问题，其中所有顶点在线到达且仅在有限时间内可用。算法必须在相关顶点同时可用时做出不可撤销的分数匹配决策。我们将经典的注水算法扩展到全在线设置。通过在线原始对偶框架，证明广义注水算法在全在线模型中竞争比为$2-\sqrt{2}\approx 0.586$且分析是紧的。为超越$2-\sqrt{2}$障碍，融入急切匹配和基于历史定价思想，得到竞争比为$0.599$的算法，证明注水算法非最优。在难度方面，进一步改进分数全在线匹配的已知上界，从Eckl等人（ORL 2021）的$0.6297$降至$0.6132$。
英文摘要
This paper studies fractional matching on general graphs in the fully online model of Huang et al. (JACM 2020), in which all vertices arrive online and remain available for only a limited time. The algorithm must make irrevocable fractional matching decisions while the relevant vertices are simultaneously available. We extend the classic Water-Filling algorithm, also known as Balance and originally introduced by Kalyanasundaram and Pruhs (TCS 2000), to the fully online setting. Using an online primal-dual framework, we prove that the generalized Water-Filling algorithm achieves a competitive ratio of $2-\sqrt{2}\approx 0.586$ in the fully online model, and that this analysis is tight. To surpass the $2-\sqrt{2}$ barrier, we incorporate the ideas of eager matching and history-based pricing into Water-Filling. We show that the resulting algorithm achieves an improved competitive ratio of $0.599$, thereby establishing that Water-Filling is not optimal in the fully online setting. On the hardness side, we further improve the known upper bound for fractional fully online matching, reducing the previous best bound of $0.6297$ due to Eckl et al. (ORL 2021) to $0.6132$.
CommentsCombined and expanded version of the fractional matching results from three conference papers: SODA 2019 (https://arxiv.org/abs/1810.07903), FOCS 2020 (https://arxiv.org/abs/2005.06311), and EC 2024 (https://arxiv.org/abs/2202.02948)
