A semi-streaming algorithm in dynamic graph streams processes any $n$-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using $O(n \cdot \mbox{polylog}(n))$ space. Semi-streaming algorithms for dynamic streams were first obtained in the seminal work of Ahn, Guha, and McGregor in 2012, alongside the introduction of the graph sketching technique, which remains the de facto way of designing algorithms in this model and a highly popular technique for designing graph algorithms in general. We settle the pass complexity of approximating maximum matchings in dynamic streams via semi-streaming algorithms by improving the state-of-the-art in both upper and lower bounds. We present a randomized sketching based semi-streaming algorithm for $O(1)$-approximation of maximum matching in dynamic streams using $O(\log\log{n})$ passes. The approximation ratio of this algorithm can be improved to $(1+\epsilon)$ for any fixed $\epsilon > 0$ even on weighted graphs using standard techniques. This exponentially improves upon several $O(\log{n})$ pass algorithms developed for this problem since the introduction of the dynamic graph streaming model. In addition, we prove that any semi-streaming algorithm (not only sketching based) for $O(1)$-approximation of maximum matching in dynamic streams requires $\Omega(\log\log{n})$ passes. This presents the first multi-pass lower bound for this problem, which is already also optimal, settling a longstanding open question in this area.

翻译：动态图流中的半流算法通过单次或多次遍历图的边插入与删除流，并使用 $O(n \cdot \mbox{polylog}(n))$ 空间来处理任意 $n$ 顶点图。动态流的半流算法最早由 Ahn、Guha 和 McGregor 在 2012 年的开创性工作中提出，同时引入了图素描技术；该技术至今仍是该模型下算法设计的实际标准方法，也是图算法设计中广受欢迎的技术。我们通过改进现有上界与下界，确定了在半流算法框架下动态流中近似最大匹配的遍历次数复杂度。我们提出一种基于随机素描的半流算法，可在动态流中以 $O(\log\log{n})$ 次遍历实现最大匹配的 $O(1)$ 近似。对于任意固定 $\epsilon > 0$，该算法的近似比可通过标准技术提升至 $(1+\epsilon)$，即使在加权图上亦然。这相较于自动态图流模型提出以来为该问题开发的多个 $O(\log{n})$ 次遍历算法实现了指数级改进。此外，我们证明任何用于动态流中最大匹配 $O(1)$ 近似的半流算法（不仅限于基于素描的方法）均需要 $\Omega(\log\log{n})$ 次遍历。这首次给出了该问题的多遍历下界，且该下界已达最优，从而解决了该领域长期存在的开放性问题。