The problem of finding a maximum size matching in a graph (known as the maximum matching problem) is one of the most classical problems in computer science. Despite a significant body of work dedicated to the study of this problem in the data stream model, the state-of-the-art single-pass semi-streaming algorithm for it is still a simple greedy algorithm that computes a maximal matching, and this way obtains 1/2-approximation. Some previous works described two/three-pass algorithms that improve over this approximation ratio by using their second and third passes to improve the above mentioned maximal matching. One contribution of this paper continuous this line of work by presenting new three-pass semi-streaming algorithms that work along these lines and obtain improved approximation ratios of 0.6111 and 0.5694 for triangle-free and general graphs, respectively. Unfortunately, a recent work (Konrad and Naidu, 2021) shows that the strategy of constructing a maximal matching in the first pass and then improving it in further passes has limitations. Additionally, this technique is unlikely to get us closer to single-pass semi-streaming algorithms obtaining a better than 1/2-approximation. Therefore, it is interesting to come up with algorithms that do something else with their first pass (we term such algorithms non-maximal-matching-first algorithms). No such algorithms are currently known (to the best of our knowledge), and the main contribution of this paper is describing such algorithms that obtain approximation ratios of 0.5384 and 0.5555 in two and three passes, respectively, for general graphs (the result for three passes improves over the previous state-of-the-art, but is worse than the result of this paper mentioned in the previous paragraph for general graphs).

中文翻译：

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最大匹配无最大匹配：一种在数据流模型中寻找最大匹配的新方法

在图中找到最大尺寸匹配的问题（称为最大匹配问题）是计算机科学中最经典的问题之一。尽管有大量的工作致力于研究数据流模型中的这个问题，但最先进的单通道半流算法仍然是一个计算最大匹配的简单贪婪算法，这方式获得 1/2 近似值。一些先前的工作描述了通过使用第二遍和第三遍来改进上述最大匹配来改进此近似比的双/三遍算法。本文的一个贡献是通过提出新的三遍半流算法来延续这条工作线，这些算法沿着这些路线工作，并为无三角形图和一般图获得 0.6111 和 0.5694 的改进近似比，分别。不幸的是，最近的一项工作（Konrad 和 Naidu，2021 年）表明，在第一次通过中构建最大匹配然后在进一步通过中改进它的策略存在局限性。此外，这种技术不太可能让我们更接近于获得比 1/2 近似更好的单通道半流算法。因此，提出在第一次通过时执行其他操作的算法很有趣（我们将此类算法称为非最大匹配优先算法）。目前还没有这样的算法（据我们所知），本文的主要贡献是描述了这样的算法，这些算法分别在两遍和三遍中获得了 0.5384 和 0.5555 的近似比，对于一般图（三遍的结果）通过改进之前的最先进技术，