In this paper, we study the weighted stochastic matching problem. Let $G=(V, E)$ be a given edge-weighted graph and let its realization $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e\in E$ independently with a known probability $p_e$. The goal in this problem is to pick a sparse subgraph $Q$ of $G$ without prior knowledge of $G$'s realization, such that the maximum weight matching among the realized edges of $Q$ (i.e. the subgraph $Q\cap \mathcal{G}$) in expectation approximates the maximum weight matching of the entire realization $\mathcal{G}$. Attaining any constant approximation ratio for this problem requires selecting a subgraph of max-degree $\Omega(1/p)$ where $p=\min_{e\in E} p_e$. On the positive side, there exists a $(1-\epsilon)$-approximation algorithm by Behnezhad and Derakhshan, albeit at the cost of max-degree having exponential dependence on $1/p$. Within the $\text{poly}(1/p)$ regime, however, the best-known algorithm achieves a $0.536$ approximation ratio due to Dughmi, Kalayci, and Patel improving over the $0.501$ approximation algorithm by Behnezhad, Farhadi, Hajiaghayi, and Reyhani. In this work, we present a 0.68 approximation algorithm with $O(1/p)$ queries per vertex, which is asymptotically tight. This is even an improvement over the best-known approximation ratio of $2/3$ for unweighted graphs within the $\text{poly}(1/p)$ regime due to Assadi and Bernstein. The $2/3$ approximation ratio is proven tight in the presence of a few correlated edges in $\mathcal{G}$, indicating that surpassing the $2/3$ barrier should rely on the independent realization of edges. Our analysis involves reducing the problem to designing a randomized matching algorithm on a given stochastic graph with some variance-bounding properties.

翻译：在本文中，我们研究加权随机匹配问题。设$G=(V, E)$为给定的边加权图，其实现$\mathcal{G}$是$G$的一个随机子图，其中每条边$e\in E$独立地以已知概率$p_e$出现。该问题的目标是在未知$G$的实现情况下，选取$G$的一个稀疏子图$Q$，使得$Q$中已实现边（即子图$Q\cap \mathcal{G}$）的最大权匹配的期望值近似于整个实现$\mathcal{G}$的最大权匹配。针对此问题，要获得任何常数近似比，需要选取一个最大度为$\Omega(1/p)$的子图，其中$p=\min_{e\in E} p_e$。在积极方面，Behnezhad和Derakhshan提出了一种$(1-\epsilon)$-近似算法，但代价是最大度对$1/p$呈指数依赖。然而，在$\text{poly}(1/p)$范围内，当前最著名的算法由Dughmi、Kalayci和Patel提出，实现了0.536的近似比，改进了Behnezhad、Farhadi、Hajiaghayi和Reyhani的0.501近似算法。在此工作中，我们提出了一种近似比为0.68的算法，每个顶点的查询次数为$O(1/p)$，这在渐近意义下是最优的。这甚至优于Assadi和Bernstein在$\text{poly}(1/p)$范围内针对无权重图所得到的最著名近似比$2/3$。当$\mathcal{G}$中存在少量相关边时，$2/3$的近似比被证明是紧的，这表明超越$2/3$的障碍应依赖于边的独立实现。我们的分析涉及将该问题归约为在给定随机图上设计一种具有方差有界性质的随机匹配算法。