We present a new approach for finding matchings in dense graphs by building on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain non-trivial albeit slight improvements over longstanding bounds for matchings in streaming and dynamic graphs. In particular, we establish the following results for $n$-vertex graphs: * A deterministic single-pass streaming algorithm that finds a $(1-o(1))$-approximate matching in $o(n^2)$ bits of space. This constitutes the first single-pass algorithm for this problem in sublinear space that improves over the $\frac{1}{2}$-approximation of the greedy algorithm. * A randomized fully dynamic algorithm that with high probability maintains a $(1-o(1))$-approximate matching in $o(n)$ worst-case update time per each edge insertion or deletion. The algorithm works even against an adaptive adversary. This is the first $o(n)$ update-time dynamic algorithm with approximation guarantee arbitrarily close to one. Given the use of regularity lemma, the improvement obtained by our algorithms over trivial bounds is only by some $(\log^*{n})^{\Theta(1)}$ factor. Nevertheless, in each case, they show that the ``right'' answer to the problem is not what is dictated by the previous bounds. Finally, in the streaming model, we also present a randomized $(1-o(1))$-approximation algorithm whose space can be upper bounded by the density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have been used extensively to prove streaming lower bounds, ours is the first to use them as an upper bound tool for designing improved streaming algorithms.

翻译：我们提出了一个在密度图形中找到匹配的新方法, 以 Szemer\'edi 所庆祝的常规性 Lemma 为基础。 这使得我们能在流和动态图形中的匹配的长期界限上取得非三进制的微小改进。 特别是, 我们为 $n 的垂直图形建立了以下结果 : * 一种确定性的单通流算法, 以 $ (1- o(1)) 在 $ (n) 2) 位位空间中找到 $ (n) 的近似匹配 。 这是用于亚线空间中这一问题的第一次单通算法, 它比 $( francial1) RS1 {1\\\\\\\\\\\\\\\\\\\\\\\ 美元 美元( $) 接近贪婪算法。 * 一个随机化的完全动态算法, 它保持$( 1- o) o- o) 最差的每次插入或删除时间值更新时间值。 算法的模型, 首先是适应于一个适应性对正统的数值。 这是第一个( IM_ dal_ max_ dal_ dal) 正确的算算法, 。 。 。 它的改进了我们的一个比正反正反正值。