  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.
