Estimating Weighted Matchings in $o(n)$ Space
  We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop an algorithm whose approximation guarantee scales by a polynomial factor.   As an application, we obtain improved estimators for weighted planar graphs and, more generally, for weighted bounded-arboricity graphs, by feeding into our reduction the recent estimators due to Esfandiari et al. (SODA 2015) and to Chitnis et al. (SODA 2016). In particular, we obtain a $(48+\epsilon)$-approximation estimator for the weight of a maximum weighted matching in planar graphs. 
Elena Grigorescu, Morteza Monemizadeh, Samson Zhou
cs.DS