Consider the following stochastic matching problem. Given a graph $G=(V, E)$, an unknown subgraph $G_p = (V, E_p)$ is realized where $E_p$ includes every edge of $E$ independently with some probability $p \in (0, 1]$. The goal is to query a sparse subgraph $H$ of $G$, such that the realized edges in $H$ include an approximate maximum matching of $G_p$. This problem has been studied extensively over the last decade due to its numerous applications in kidney exchange, online dating, and online labor markets. For any fixed $\epsilon > 0$, [BDH STOC'20] showed that any graph $G$ has a subgraph $H$ with $\text{quasipoly}(1/p) = (1/p)^{\text{poly}(\log(1/p))}$ maximum degree, achieving a $(1-\epsilon)$-approximation. A major open question is the best approximation achievable with $\text{poly}(1/p)$-degree subgraphs. A long line of work has progressively improved the approximation in the $\text{poly}(1/p)$-degree regime from .5 [BDH+ EC'15] to .501 [AKL EC'17], .656 [BHFR SODA'19], .666 [AB SOSA'19], .731 [BBD SODA'22] (bipartite graphs), and most recently to .68 [DS '24]. In this work, we show that a $\text{poly}(1/p)$-degree subgraph can obtain a $(1-\epsilon)$-approximation for any desirably small fixed $\epsilon > 0$, achieving the best of both worlds. Beyond its quantitative improvement, a key conceptual contribution of our work is to connect local computation algorithms (LCAs) to the stochastic matching problem for the first time. While prior work on LCAs mainly focuses on their out-queries (the number of vertices probed to produce the output of a given vertex), our analysis also bounds the in-queries (the number of vertices that probe a given vertex). We prove that the outputs of LCAs with bounded in- and out-queries (in-n-out LCAs for short) have limited correlation, a property that our analysis crucially relies on and might find applications beyond stochastic matchings.

翻译：考虑以下随机匹配问题。给定图$G=(V, E)$，一个未知子图$G_p = (V, E_p)$被实现，其中$E_p$以概率$p \in (0, 1]$独立包含$E$的每条边。目标是在$G$中查询一个稀疏子图$H$，使得$H$中实现的边包含$G_p$的近似最大匹配。该问题在过去十年中因其在肾脏交换、在线约会和在线劳动力市场中的广泛应用而得到深入研究。对于任意固定$\epsilon > 0$，[BDH STOC'20]证明任意图$G$都存在最大度为$\text{quasipoly}(1/p) = (1/p)^{\text{poly}(\log(1/p))}$的子图$H$，能够实现$(1-\epsilon)$-近似。一个主要的开放问题是：在$\text{poly}(1/p)$-度子图中可实现的最佳近似比是多少？一系列研究工作逐步将$\text{poly}(1/p)$-度机制下的近似比从0.5 [BDH+ EC'15]提升至0.501 [AKL EC'17]、0.656 [BHFR SODA'19]、0.666 [AB SOSA'19]、0.731 [BBD SODA'22]（二分图），最近达到0.68 [DS '24]。本研究表明，对于任意期望小的固定$\epsilon > 0$，$\text{poly}(1/p)$-度子图能够实现$(1-\epsilon)$-近似，从而达成两全其美的结果。除了定量改进外，本研究的关键概念贡献在于首次将局部计算算法（LCAs）与随机匹配问题建立联系。虽然先前关于LCAs的研究主要关注其外向查询（为生成给定顶点的输出而探测的顶点数量），我们的分析同时界定了内向查询（探测给定顶点的顶点数量）。我们证明具有有界内外查询的LCAs（简称in-n-out LCAs）的输出具有有限相关性，这一性质是我们分析的关键依据，并可能在随机匹配之外的其他领域得到应用。