The stochastic block model (SBM) is a generalization of the Erd\H{o}s--R\'enyi model of random graphs that describes the interaction of a finite number of distinct communities. In sparse Erd\H{o}s--R\'enyi graphs, it is known that a linear-time algorithm of Karp and Sipser achieves near-optimal matching sizes asymptotically almost surely, giving a law-of-large numbers for the matching sizes of such graphs in terms of solutions to an ODE. We provide an extension of this analysis, identifying broad ranges of stochastic block model parameters for which the Karp--Sipser algorithm achieves near-optimal matching sizes, but demonstrating that it cannot perform optimally on general SBM instances. We also consider the problem of constructing a matching online, in which the vertices of one half of a bipartite stochastic block model arrive one-at-a-time, and must be matched as they arrive. We show that the competitive ratio lower bound of 0.837 found by Mastin and Jaillet for the Erd\H{o}s--R\'enyi case is tight whenever the expected degrees in all communities are equal. We propose several linear-time algorithms for online matching in the general stochastic block model, but prove that despite very good experimental performance, none of these achieve online asymptotic optimality.

翻译：随机块模型（SBM）是随机图 Erdős–Rényi 模型的推广，描述了有限个不同社区之间的相互作用。在稀疏 Erdős–Rényi 图中，已知 Karp-Sipser 的线性时间算法渐近几乎必然达到接近最优的匹配规模，从而以常微分方程的解形式给出了此类图匹配规模的强大数定律。我们将此分析进行扩展，识别出随机块模型参数的广泛范围，在此范围内 Karp-Sipser 算法可实现接近最优的匹配规模，但证明该算法无法在一般 SBM 实例上达到最优性能。我们同时考虑了在线匹配的构建问题，其中二分随机块模型一侧的顶点逐个到达，且必须在到达时完成匹配。我们证明当所有社区的期望度相等时，Mastin 和 Jaillet 针对 Erdős–Rényi 情形得到的 0.837 竞争比下界是紧的。我们提出了多种适用于一般随机块模型在线匹配的线性时间算法，但证明尽管实验性能优异，这些算法均无法实现渐近在线最优性。