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.

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