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:我们展示了一种完全动态的算法, 用于维持$(1 ⁇ - epsilon) $- 近似 emph{ 大小}, 使图表与 $n of- omega- omega- ⁇ - epsilon} (1)} 更新时间 。 这是长期 $O (n) 更新时间的首次多线性改进, 可以通过定期重计获得。 因此, 我们解决了动态图形算法文献中一个主要的未解答问题的价值版本( 例如, 参见 [ Gupta 和 Peng FOCS' 13], [ Bernstein和 Stein SODA'16], [Behnezhad 和 Khanna SODA' 22] 。 我们的关键技术组件是 $(1, 1,\\\\ epsilon n) $- pappoint imal ad ad ad adly potrainal y potings polar polar plographengraphic grapht. glogn.