  We design a generic method for reducing the task of finding weighted
matchings to that of finding short augmenting paths in unweighted graphs. This
method enables us to provide efficient implementations for approximating
weighted matchings in the streaming model and in the massively parallel
computation (MPC) model.
  In the context of streaming with random edge arrivals, our techniques yield a
$(1/2+c)$-approximation algorithm thus breaking the natural barrier of $1/2$.
For multi-pass streaming and the MPC model, we show that any algorithm
computing a $(1-\delta)$-approximate unweighted matching in bipartite graphs
can be translated into an algorithm that computes a
$(1-\varepsilon(\delta))$-approximate maximum weighted matching. Furthermore,
this translation incurs only a constant factor (that depends on $\varepsilon>
0$) overhead in the complexity. Instantiating this with the current best
multi-pass streaming and MPC algorithms for unweighted matchings yields the
following results for maximum weighted matchings:
  * A $(1-\varepsilon)$-approximation streaming algorithm that uses
$O_\varepsilon(1)$ passes and $O_\varepsilon(n\, \text{poly} (\log n))$ memory.
This is the first $(1-\varepsilon)$-approximation streaming algorithm for
weighted matchings that uses a constant number of passes (only depending on
$\varepsilon$).
  * A $(1 - \varepsilon)$-approximation algorithm in the MPC model that uses
$O_\varepsilon(\log \log n)$ rounds, $O(m/n)$ machines per round, and
$O_\varepsilon(n\, \text{poly}(\log n))$ memory per machine. This improves upon
the previous best approximation guarantee of $(1/2-\varepsilon)$ for weighted
graphs.
