We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than $2$. Specifically, we obtain a $1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$ approximation in bipartite graphs and a $1.973+\epsilon$ approximation in general graphs. We thus answer in the affirmative the major open question first posed in the influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms also work against an adaptive adversary and guarantee worst-case polylog update time, both w.h.p. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

翻译：我们提出了具有对数级更新时间的动态算法，用于估计经历边插入和删除的图的最大匹配大小，其近似比严格优于$2$。具体地，我们在二分图中获得了$1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$的近似比，在一般图中获得了$1.973+\epsilon$的近似比。因此，我们肯定地回答了Onak和Rubinfeld（STOC'10）有影响力的工作中首次提出并在动态图算法文献中反复提及的主要开放问题。我们的随机化算法也适用于自适应对手，并能保证最坏情况下的对数级更新时间（两者均以高概率成立）。我们的算法基于在动态设置中模拟新的两遍流式匹配算法。我们的关键新思路是以白盒方式调用Behnezhad（FOCS'21）最近的亚线性时间匹配算法，以高效模拟流式算法的第二遍处理，同时绕过众所周知的顶点更新瓶颈。