We study the problem of computing an approximate maximum cardinality matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, the edges of the input graph \(G = (V,E)\) are given as a stream \(e_1, \ldots , e_m\), and the algorithm is allowed to make a single pass over this stream while using \(O(n\text {polylog}(n))\) space (\(m = |E|\) and \(n = |V|\)). If the order of edges is adversarial, a simple single-pass greedy algorithm yields a 1/2-approximation in O(n) space; achieving a better approximation in adversarial streams remains an elusive open question. A line of recent work shows that one can improve upon the 1/2-approximation if the edges of the stream arrive in a random order. The state of the art for this model is two-fold: Assadi et al. [SODA 2019] show how to compute a \(\frac{2}{3}\) \((\sim .66)\)-approximate matching, but the space requirement is \(O(n^{1.5}\text {polylog}(n))\). Very recently, Farhadi et al. [SODA 2020] presented an algorithm with the desired space usage of \(O(n\text {polylog}(n))\), but a worse approximation ratio of \(\frac{6}{11}\) \((\sim .545)\), or \(\frac{3}{5}\) \((=.6)\) in bipartite graphs. In this paper, we present an algorithm that computes a \(\frac{2}{3}(\sim .66)\)-approximate matching using only \(O(n\log (n))\) space, improving upon both results above. We also note that for adversarial streams, a lower bound of Kapralov [SODA 2013] shows that any algorithm that achieves a \(1-\frac{1}{e}\)(\(\sim .63\))-approximation requires \((n^{1+\Omega (1/\log \log (n))})\) space; recent follow-up work by the same author improved this lower bound to \(1+\ln (2) \sim .59\) [SODA 2021]. As a consequence, both our result and the earlier result of Farhadi et al. prove that the problem of computing a maximum matching is strictly easier in random-order streams than in adversarial ones.

我们研究当边缘以随机顺序到达时在半流模型中计算近似最大基数匹配的问题。在半流模型中，输入图\(G = (V,E)\)的边作为流\(e_1,\ldots,e_m\)给出，并且允许算法进行单遍在此流上，同时使用\(O(n\text {polylog}(n))\)空间（\(m = |E|\)和\(n = |V|\)）。如果边的顺序是对抗性的，则简单的单遍贪婪算法会在O ( n ) 空间中产生 1/2 近似值；在对抗性流中实现更好的近似仍然是一个难以捉摸的悬而未决的问题。最近的一系列工作表明，如果流的边缘以随机顺序到达，则可以改进 1/2 近似。该模型的最新技术有两个方面：Assadi 等人。[SODA 2019] 展示如何计算\(\frac{2}{3}\) \((\sim .66)\)近似匹配，但空间要求为\(O(n^{1.5}\)文本 {polylog}(n))\)。最近，Farhadi 等人。[SODA 2020] 提出了一种算法，其所需的空间使用量为\(O(n\text {polylog}(n))\)，但近似率更差为\(\frac{6}{11}\) \( (\sim .545)\)或二分图中的\(\frac{3}{5}\) \((=.6)\) 。在本文中，我们提出了一种仅使用\(O(n\log (n))\)空间计算\(\frac{2}{3}(\sim .66)\)近似匹配的算法，改进了根据上述两个结果。我们还注意到，对于对抗流，Kapralov [SODA 2013] 的下界表明任何实现\(1-\frac{1}{e}\) ( \(\sim .63\) ) 近似的算法需要\((n^{1+\Omega(1/\log \log (n))})\)空间；同一作者最近的后续工作将这个下限改进为\(1+\ln (2) \sim .59\) [SODA 2021]。因此，我们的结果和 Farhadi 等人的早期结果。证明计算最大匹配的问题在随机顺序流中比在对抗性流中更容易。