We study maximum matchings in fully dynamic graphs, which are graphs that undergo both edge insertions and deletions. Our focus is on algorithms that estimate the size of maximum matching after each update while spending a small time. An important question studied extensively is the best approximation achievable via algorithms that only spend $\text{poly}(\log n)$ time per update, where $n$ is the number of vertices. The current best bound is a $(1/2+\varepsilon_0)$-approximation for a small constant $\varepsilon_0 > 0$, due to recent works of Behnezhad [SODA'23] ($\varepsilon_0 \sim 0.001$) and Bhattacharya, Kiss, Saranurak, Wajc [SODA'23] ($\varepsilon_0 \sim 0.006$) who broke the long-standing 1/2-approximation barrier. These works also showed that for any fixed $\varepsilon > 0$, the approximation can be further improved to $(2-\sqrt{2}-\varepsilon) \sim .585$ for bipartite graphs, leaving a huge gap between general and bipartite graphs. In this work, we close this gap. We show that for any fixed $\varepsilon > 0$, a $(2-\sqrt{2}-\varepsilon)$ approximation can be maintained in $\text{poly}(\log n)$ time per update even in general graphs. Our techniques also lead to the same approximation for general graphs in two passes of the semi-streaming setting, removing a similar gap in that setting.

翻译：我们研究完全动态图中的最大匹配问题，这类图会经历边的插入和删除操作。我们重点关注每次更新后估算最大匹配大小且耗时较短的算法。一个被广泛研究的重要问题是：在每次更新仅花费$\text{poly}(\log n)$时间的算法中，最佳可实现近似比是多少（其中$n$为顶点数）。当前最优结果为$(1/2+\varepsilon_0)$近似（$\varepsilon_0 > 0$为小常数），这源于Behnezhad [SODA'23]（$\varepsilon_0 \sim 0.001$）与Bhattacharya、Kiss、Saranurak、Wajc [SODA'23]（$\varepsilon_0 \sim 0.006$）的最新工作，他们突破了长期存在的1/2近似瓶颈。这些工作还表明，对任意固定$\varepsilon > 0$，二分图的近似比可进一步提升至$(2-\sqrt{2}-\varepsilon) \sim .585$，但一般图与二分图之间仍存在巨大差距。本文填补了这一空白。我们证明：对任意固定$\varepsilon > 0$，即使在一般图中，也可在每次更新$\text{poly}(\log n)$时间内维持$(2-\sqrt{2}-\varepsilon)$近似。我们的技术还使得半流式（semi-streaming）设定下一般图的两遍扫描能达到相同近似比，消除了该设定中的类似差距。