  We consider the {\em stochastic matching} problem. An edge-weighted general
(i.e., not necessarily bipartite) graph $G(V, E)$ is given in the input, where
each edge in $E$ is {\em realized} independently with probability $p$; the
realization is initially unknown, however, we are able to {\em query} the edges
to determine whether they are realized. The goal is to query only a small
number of edges to find a {\em realized matching} that is sufficiently close to
the maximum matching among all realized edges. This problem has received a
considerable attention during the past decade due to its numerous real-world
applications in kidney-exchange, matchmaking services, online labor markets,
and advertisements.
  Our main result is an {\em adaptive} algorithm that for any arbitrarily small
$\epsilon > 0$, finds a $(1-\epsilon)$-approximation in expectation, by
querying only $O(1)$ edges per vertex. We further show that our approach leads
to a $(1/2-\epsilon)$-approximate {\em non-adaptive} algorithm that also
queries only $O(1)$ edges per vertex. Prior to our work, no nontrivial
approximation was known for weighted graphs using a constant per-vertex budget.
The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and
Yamaguchi [SODA 2018] achieves a $(1-\epsilon)$-approximation (resp.
$(1/2-\epsilon)$-approximation) by querying up to $O(w\log{n})$ edges per
vertex where $w$ denotes the maximum integer edge-weight. Our result is a
substantial improvement over this bound and has an appealing message: No matter
what the structure of the input graph is, one can get arbitrarily close to the
optimum solution by querying only a constant number of edges per vertex.
  To obtain our results, we introduce novel properties of a generalization of
{\em augmenting paths} to weighted matchings that may be of independent
interest.
