In the classic online stochastic matching proposed by Feldman et al. (FOCS 2009), there is a known bipartite type-graph, where one side of the graph is given offline. Upon the arrival of each online vertex, its type is sampled independently and identically from the other side of the type-graph. This model has been extensively studied over the past decade, yielding a rich body of theoretical results. In this paper, we initiate the study of an edge arrival model for online stochastic matching. In our model, the online edges are sampled independently and identically (KIID) from a known type-graph, which need not be bipartite. We first show that the Greedy algorithm cannot achieve a competitive ratio strictly better than $0.5$ while the Suggested Matching algorithm has a competitive ratio of $1-1/e$ under the assumption of integral arrival rates, matching its performance in the one-sided vertex arrival model. We then propose a two-stage algorithm that combines Greedy and Suggested Matching, and show that its competitive ratio is strictly higher than $1-1/e$ for integral arrival rates. While our algorithm is simple, its analysis is intricate and builds upon the Natural LP, which has been proven very powerful in vertex arrival models. Our result reveals that even in the more challenging edge arrival setting for general graphs, competitive ratios better than $1-1/e$ are still possible, given the known distributions.

翻译：在Feldman等人（FOCS 2009）提出的经典在线随机匹配问题中，存在一个已知的二部类型图，其中图的一侧离线给定。每个在线顶点到达时，其类型独立同分布地从类型图的另一侧采样。该模型在过去十年中得到了广泛研究，积累了丰富的理论成果。本文首次研究在线随机匹配的边到达模型。在我们的模型中，在线边独立同分布（KIID）地从已知类型图中采样，该图无需是二部图。我们首先证明贪心算法的竞争比严格低于0.5，而建议匹配算法在整数到达率假设下达到1-1/e的竞争比，与单侧顶点到达模型中的表现一致。随后，我们提出一种结合贪心与建议匹配的两阶段算法，并证明其在整数到达率下的竞争比严格高于1-1/e。尽管算法本身简洁，其分析却颇为复杂，且建立在自然线性规划（Natural LP）基础上——该工具在顶点到达模型中已被证明极为有效。我们的结果表明，即使在一般图的更具挑战性的边到达设定中，给定已知分布的情况下，仍可能实现优于1-1/e的竞争比。