  This paper presents an $O(\log\log \bar{d})$ round massively parallel
algorithm for $1+\epsilon$ approximation of maximum weighted $b$-matchings,
using near-linear memory per machine. Here $\bar{d}$ denotes the average degree
in the graph and $\epsilon$ is an arbitrarily small positive constant. Recall
that $b$-matching is the natural and well-studied generalization of the
matching problem where different vertices are allowed to have multiple (and
differing number of) incident edges in the matching. Concretely, each vertex
$v$ is given a positive integer budget $b_v$ and it can have up to $b_v$
incident edges in the matching. Previously, there were known algorithms with
round complexity $O(\log\log n)$, or $O(\log\log \Delta)$ where $\Delta$
denotes maximum degree, for $1+\epsilon$ approximation of weighted matching and
for maximal matching [Czumaj et al., STOC'18, Ghaffari et al. PODC'18; Assadi
et al. SODA'19; Behnezhad et al. FOCS'19; Gamlath et al. PODC'19], but these
algorithms do not extend to the more general $b$-matching problem.
