In a seminal paper on finding large matchings in sparse random graphs, Karp and Sipser proposed two algorithms for this task. The second algorithm has been intensely studied, but due to technical difficulties, the first algorithm has received less attention. Empirical results by Karp and Sipser suggest that the first algorithm is superior. In this paper we show that this is indeed the case, at least for random k‐regular graphs. We show that w.h.p. the first algorithm will find a matching of size in a random k‐regular graph, k = O(1). We also show that the algorithm can be adapted to find a maximum matching in O(n) time w.h.p. This is to be compared with O(n^3/2) time for the worst‐case.

中文翻译：

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在线性期望时间内在随机正则图中找到最大匹配

在关于在稀疏随机图中找到大匹配项的开创性论文中，Karp和Sipser提出了两种算法来完成此任务。已经对第二种算法进行了深入研究，但是由于技术上的困难，第一种算法受到的关注较少。Karp和Sipser的经验结果表明，第一种算法是更好的。在本文中，我们证明确实如此，至少对于随机k正则图而言。我们证明了第一种算法将在随机k正则图中k  =  O（1）处找到大小的匹配项。我们还表明，该算法可以适用于在O（n）中找到最大匹配项time whp这将与最坏情况下的O（n ^3/2）时间进行比较。