The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, however, we still have a far more limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of $n^{1+\Omega(1)}$ per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm a variant of which was conjectured to work by Czumaj et al. [STOC'18]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among some other results, leads to an $O(\log \log \Delta)$ round algorithm for maximal matching with $O(n)$ space (or even mildly sublinear in $n$ using standard techniques). As an immediate corollary, we get a $2$ approximate minimum vertex cover in essentially the same rounds and space. This is the best possible approximation factor under standard assumptions, culminating a long line of research. It also leads to an improved $O(\log\log \Delta)$ round algorithm for $1 + \varepsilon$ approximate matching. All these results can also be implemented in the congested clique model within the same number of rounds.

翻译：大规模并行计算（MPC）模型中近似匹配的研究近期取得了突破性进展。然而，尽管取得了这些进展，我们对最大匹配——这一并行与分布式计算核心问题之一——的理解仍然远远不足。所有已知的最大匹配MPC算法要么需要被认为是低效的多对数时间，要么要求每台机器具有严格超线性的空间$n^{1+\Omega(1)}$。在本工作中，我们通过一种极其简单算法（Czumaj等人[STOC'18]曾猜想其变体有效）的新颖分析填补了这一空白。该算法对图进行边采样，随机划分顶点，并在每个划分内寻找随机贪心最大匹配。我们证明该算法能大幅降低顶点度数。结合其他若干结果，这导出了一个最大匹配的$O(\log \log \Delta)$轮算法，且仅需$O(n)$空间（甚至可利用标准技术实现轻微亚线性空间）。作为直接推论，我们能在基本相同的轮数和空间内得到2-近似最小顶点覆盖。这是在标准假设下可能达到的最佳近似因子，终结了长期的研究脉络。该结果还导出了$1 + \varepsilon$近似匹配的改进$O(\log\log \Delta)$轮算法。所有这些结果同样可在拥塞团模型中以相同轮数实现。