稀疏随机块模型中的匹配算法
Matching Algorithms in the Sparse Stochastic Block Model
AI总结：
本文研究稀疏 SBM 中的离线与在线匹配，分析 Karp–Sipser 的近优参数范围，并提出多种线性时间在线算法及 0.837 紧下界。
摘要：
随机块模型（SBM）是 Erdős–Rényi 随机图模型的推广，用于描述有限数量不同社区之间的交互。在稀疏 Erdős–Rényi 图中，已知 Karp 和 Sipser 提出的线性时间算法能以渐近几乎必然的方式取得接近最优的匹配规模，并通过一个 ODE 的解给出了此类图匹配规模的大数定律。我们扩展了这一分析，确定了 Karp–Sipser 算法能够取得接近最优匹配规模的广泛随机块模型参数范围，同时也证明它在一般 SBM 实例上无法达到最优。我们还考虑在线构造匹配的问题：在二分随机块模型中，一侧顶点逐个到达，并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 在 Erdős–Rényi 情形下得到的 0.837 竞争比下界是紧的。我们为一般随机块模型中的在线匹配提出了若干线性时间算法；尽管这些算法在实验中表现非常好，但我们证明它们都不能实现在线渐近最优。
英文摘要：
The stochastic block model (SBM) is a generalization of the Erdős--Rényi model of random graphs that describes the interaction of a finite number of distinct communities. In sparse Erdős--Rényi 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 Erdős--Rényi 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
