$ $We study the $d$-Uniform Hypergraph Matching ($d$-UHM) problem: given an $n$-vertex hypergraph $G$ where every hyperedge is of size $d$, find a maximum cardinality set of disjoint hyperedges. For $d\geq3$, the problem of finding the maximum matching is NP-complete, and was one of Karp's 21 $\mathcal{NP}$-complete problems. In this paper we are interested in the problem of finding matchings in hypergraphs in the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In this model, we present the first three parallel algorithms for $d$-Uniform Hypergraph Matching, and we analyse them in terms of resources such as memory usage, rounds of communication needed, and approximation ratio. The highlights include: $\bullet$ A $O(\log n)$-round $d$-approximation algorithm that uses $O(nd)$ space per machine. $\bullet$ A $3$-round, $O(d^2)$-approximation algorithm that uses $\tilde{O}(\sqrt{nm})$ space per machine. $\bullet$ A $3$-round algorithm that computes a subgraph containing a $(d-1+\frac{1}{d})^2$-approximation, using $\tilde{O}(\sqrt{nm})$ space per machine for linear hypergraphs, and $\tilde{O}(n\sqrt{nm})$ in general.

中文翻译：

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用于超图中匹配的分布式算法

$ $ 我们研究 $d$-Uniform Hypergraph Matching ($d$-UHM) 问题：给定 $n$-顶点超图 $G$，其中每个超边的大小为 $d$，找到不相交超边的最大基数集. 对于 $d\geq3$，寻找最大匹配的问题是 NP-complete，是 Karp 的 21 个 $\mathcal{NP}$-complete 问题之一。在本文中，我们对在大规模并行计算 (MPC) 模型中寻找超图中匹配的问题感兴趣，该模型是 MapReduce 式计算的常见抽象。在该模型中，我们展示了 $d$-Uniform Hypergraph Matching 的前三种并行算法，并根据内存使用情况、所需的通信轮数和逼近率等资源对其进行了分析。亮点包括：$\bullet$ $O(\log n)$-round $d$-近似算法，每台机器使用 $O(nd)$ 空间。$\bullet$ 一个 $3$-round，$O(d^2)$-近似算法，每台机器使用 $\tilde{O}(\sqrt{nm})$ 空间。$\bullet$ 一个 $3$-round 算法，计算一个包含 $(d-1+\frac{1}{d})^2$-近似值的子图，使用 $\tilde{O}(\sqrt{nm} )$ 每台机器的空间用于线性超图，以及 $\tilde{O}(n\sqrt{nm})$ 一般。