Suppose that we are given an arbitrary graph $G=(V, E)$ and know that each edge in $E$ is going to be realized independently with some probability $p$. The goal in the stochastic matching problem is to pick a sparse subgraph $Q$ of $G$ such that the realized edges in $Q$, in expectation, include a matching that is approximately as large as the maximum matching among the realized edges of $G$. The maximum degree of $Q$ can depend on $p$, but not on the size of $G$. This problem has been subject to extensive studies over the years and the approximation factor has been improved from $0.5$ to $0.5001$ to $0.6568$ and eventually to $2/3$. In this work, we analyze a natural sampling-based algorithm and show that it can obtain all the way up to $(1-\epsilon)$ approximation, for any constant $\epsilon > 0$. A key and of possible independent interest component of our analysis is an algorithm that constructs a matching on a stochastic graph, which among some other important properties, guarantees that each vertex is matched independently from the vertices that are sufficiently far. This allows us to bypass a previously known barrier towards achieving $(1-\epsilon)$ approximation based on existence of dense Ruzsa-Szemer\'edi graphs.

中文翻译：

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具有少量查询的随机匹配：$(1-\varepsilon)$ 近似

假设我们给定了一个任意图 $G=(V, E)$ 并且知道 $E$ 中的每条边将以某种概率 $p$ 独立实现。随机匹配问题的目标是选择 $G$ 的稀疏子图 $Q$，使得 $Q$ 中的已实现边在预期中包含一个匹配，该匹配大约与已实现边之间的最大匹配一样大$G$。$Q$ 的最大度数可以取决于 $p$，但不取决于 $G$ 的大小。多年来，对这个问题进行了广泛的研究，近似因子已从 0.5 美元提高到 0.5001 美元，再到 0.6568 美元，最终提高到 2/3 美元。在这项工作中，我们分析了一种基于自然采样的算法，并表明对于任何常数 $\epsilon > 0$，它可以一直获得 $(1-\epsilon)$ 的近似值。我们分析的一个关键和可能的独立兴趣组件是一种在随机图上构建匹配的算法，该算法在一些其他重要属性中保证每个顶点与足够远的顶点独立匹配。这使我们能够绕过先前已知的基于密集 Ruzsa-Szemer\'edi 图的存在来实现 $(1-\epsilon)$ 近似的障碍。