We show a fully dynamic algorithm for maintaining $(1+\epsilon)$-approximate \emph{size} of maximum matching of the graph with $n$ vertices and $m$ edges using $m^{0.5-\Omega_{\epsilon}(1)}$ update time. This is the first polynomial improvement over the long-standing $O(n)$ update time, which can be trivially obtained by periodic recomputation. Thus, we resolve the value version of a major open question of the dynamic graph algorithms literature (see, e.g., [Gupta and Peng FOCS'13], [Bernstein and Stein SODA'16],[Behnezhad and Khanna SODA'22]). Our key technical component is the first sublinear algorithm for $(1,\epsilon n)$-approximate maximum matching with sublinear running time on dense graphs. All previous algorithms suffered a multiplicative approximation factor of at least $1.499$ or assumed that the graph has a very small maximum degree.

翻译：我们提出一种全动态算法，用于维护包含 $n$ 个顶点和 $m$ 条边的图的最大匹配的 $(1+\epsilon)$-近似大小，其更新时间复杂度为 $m^{0.5-\Omega_{\epsilon}(1)}$。这是对长期存在的 $O(n)$ 更新时间（可通过周期性重计算平凡实现）的首次多项式改进。因此，我们解决了动态图算法文献中一个重大公开问题的“数值版本”（例如，参见 [Gupta and Peng FOCS'13]、[Bernstein and Stein SODA'16]、[Behnezhad and Khanna SODA'22]）。我们的关键技术贡献是第一个在稠密图上具有次线性运行时间的 $(1,\epsilon n)$-近似最大匹配的次线性算法。此前所有算法要么遭受至少 $1.499$ 的乘法近似因子，要么假设图的极大度数非常小。