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).
链接流是三元组$(t, u, v)$的集合，其指示在时间t在u和v之间发生交互。我们推广的经典概念的图中的团，这样的链接流：对于一个给定的$Δ$，一个$Δ$ -团是一组节点和一个时间间隔，使得在这个集合中的所有节点对在每个子间隔的持续时间$Δ$至少交互一次。我们提出了一个算法来枚举所有最大的（在节点或时间间隔方面）的链接流的集团，并说明其在现实世界中的接触轨迹的实际意义。
我们认为经典的最大和最大独立集问题的三种模型的图流：在边缘到达模型中，我们看到一个流的边缘，共同定义一个图，这个模型已经很好地研究了各种问题。我们首先表明，一个一遍流算法找到一个最大的独立集的空间复杂度是二次的（即，我们必须存储所有的边缘）。我们进一步表明，问题并没有变得容易得多，如果我们只需要近似极大。在“显式”顶点流模型中，输入流是组成图的顶点序列，其中每个顶点都与连接到先前到达的顶点的关联边一起到达。在这种设置中，各种图形问题需要比边缘到达流少得多的空间来解决。我们证明了显式顶点流上的最大独立集（MIS）的每个一遍$c$ -近似算法都需要空间$Ω(\frac{n^2}{c^7})$，其中$n$是输入图的顶点数，并且已经知道空间$\tildeΘ(\frac{n^2}{c^2})$在边到达模型中是必要和充分的（Halldórsson et al. 2012）。因此，MIS问题在显式顶点到达顺序假设下并不容易解决。我们的结果证明通过减少到一个新的多方通信问题密切相关的指针跳跃。在“隐式”顶点流模型中，输入流由一系列对象组成，每个顶点一个。该算法配备了一个功能，可以将一对对象映射到边缘的存在或不存在，从而定义图形。该模型捕获例如几何求交图，诸如单位圆盘图。我们的最终结果包括几个改进的上界和下界球相交图，在显式和隐式流。
