  We provide CONGEST model algorithms for approximating minimum weighted vertex
cover and the maximum weighted matching. For bipartite graphs, we show that a
$(1+\varepsilon)$-approximate weighted vertex cover can be computed
deterministically in polylogarithmic time. This generalizes a corresponding
result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS
'20]. Moreover, we show that in general weighted graph families that are closed
under taking subgraphs and in which we can compute an independent set of weight
at least a $\lambda$-fraction of the total weight, one can compute a
$(2-2\lambda +\varepsilon)$-approximate weighted vertex cover in
polylogarithmic time in the CONGEST model. Our result in particular implies
that in graphs of arboricity $a$, one can compute a
$(2-1/a+\varepsilon)$-approximate weighted vertex cover.
  For maximum weighted matchings, we show that a $(1-\varepsilon)$-approximate
solution can be computed deterministically in polylogarithmic CONGEST rounds
(for constant $\varepsilon$). We also provide a more efficient randomized
algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie;
SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the
unweighted case.
  Finally, we show that even in the LOCAL model and in bipartite graphs of
degree $\leq 3$, if $\varepsilon<\varepsilon_0$ for some constant
$\varepsilon_0>0$, then computing a $(1+\varepsilon)$-approximation for the
unweighted minimum vertex cover problem requires $\Omega\big(\frac{\log
n}{\varepsilon}\big)$ rounds. This generalizes aresult of [G\"o\"os, Suomela;
DISC '12], who showed that computing a $(1+\varepsilon_0)$-approximation in
such graphs requires $\Omega(\log n)$ rounds.
