  We consider the following stochastic matching problem on both weighted and
unweighted graphs: A graph $G(V, E)$ along with a parameter $p \in (0, 1)$ is
given in the input. Each edge of $G$ is realized independently with probability
$p$. The goal is to select a degree bounded (dependent only on $p$) subgraph
$H$ of $G$ such that the expected size/weight of maximum realized matching of
$H$ is close to that of $G$.
  This model of stochastic matching has attracted significant attention over
the recent years due to its various applications. The most fundamental open
question is the best approximation factor achievable for such algorithms that,
in the literature, are referred to as non-adaptive algorithms. Prior work has
identified breaking (near) half-approximation as a barrier for both weighted
and unweighted graphs. Our main results are as follows:
  -- We analyze a simple and clean algorithm and show that for unweighted
graphs, it finds an (almost) $4\sqrt{2}-5$ ($\approx 0.6568$) approximation by
querying $O(\frac{\log (1/p)}{p})$ edges per vertex. This improves over the
state-of-the-art $0.5001$ approximate algorithm of Assadi et al. [EC'17].
  -- We show that the same algorithm achieves a $0.501$ approximation for
weighted graphs by querying $O(\frac{\log (1/p)}{p})$ edges per vertex. This is
the first algorithm to break $0.5$ approximation barrier for weighted graphs.
It also improves the per-vertex queries of the state-of-the-art by Yamaguchi
and Maehara [SODA'18] and Behnezhad and Reyhani [EC'18].
  Our algorithms are fundamentally different from prior works, yet are very
simple and natural. For the analysis, we introduce a number of procedures that
construct heavy fractional matchings. We consider the new algorithms and our
analytical tools to be the main contributions of this paper.
