The online stochastic matching problem was introduced by [FMMM09], together with the $(1-\frac1e)$-competitive Suggested Matching algorithm. In the most general edge-weighted setting, this ratio has not been improved for more than one decade, until recently [Yan24] beat the $1-\frac1e$ bound and [QFZW23] further improved it to $0.650$. Both works measure the online competitiveness against the offline LP relaxation introduced by Jaillet and Lu [JL14]. The same LP has also played an important role in other settings as it is a natural choice for two-choice online algorithms. In this paper, we prove an upper bound of $0.663$ and a lower bound of $0.662$ for edge-weighted online stochastic matching under Jaillet-Lu LP. We propose a simple hard instance and identify the optimal online algorithm for this specific instance which has a competitive ratio of $<0.663$. Despite the simplicity of the instance, we then show that a near-optimal algorithm for it, which has a competitive ratio of $>0.662$, can be generalized to work on all instances without any loss. As our algorithm is generalized from a real near-optimal algorithm instead of manually combining trivial strategies, it has two natural advantages compared with previous works: (1) its matching strategy varies from time to time; (2) it utilizes global information about offline vertices. On the other hand, the upper bound suggests that more powerful LPs and multiple-choice strategies are needed if we want to further improve the ratio by $>0.001$. In addition to our main result, we also generalize the asymptotic equivalence between the Poisson arrival model and the original online stochastic matching established by [HS21], removing the requirement of approximate monotonicity for the online algorithm.

翻译：摘要：在线随机匹配问题由[FMMM09]提出，并给出了$(1-\frac1e)$竞争比的建议匹配算法。在最具一般性的边加权设置下，该比率在十多年间未被改进，直至近期[Yan24]突破了$1-\frac1e$界限，而[QFZW23]进一步将其提升至$0.650$。这两项工作均以Jaillet与Lu[JL14]提出的离线线性规划松弛作为衡量在线竞争比的基准。该线性规划在其他场景中也扮演着重要角色，因其是双选择在线算法的自然选择。本文证明了Jaillet-Lu线性规划下边加权在线随机匹配的上界为$0.663$，下界为$0.662$。我们构造了一个简洁的困难实例，并确定了该实例下竞争比<$0.663$的最优在线算法。进一步地，我们证明该实例的近似最优算法（竞争比>$0.662$）可无损推广至所有实例。由于该算法源于真实的近似最优策略而非人工组合平凡策略，它相较以往工作具有两个自然优势：(1)其匹配策略随时间动态变化；(2)能利用离线节点的全局信息。另一方面，上界结果表明，若需将竞争比再提升$>0.001$，则需要更强大的线性规划及多选择策略。除主要结果外，我们还推广了[HS21]建立的泊松到达模型与原在线随机匹配之间的渐近等价性，去除了对在线算法近似单调性的要求。