  We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
$r$. Our main result is a deterministic algorithm to generate a matching which
is an $O(r)$-approximation to the maximum weight matching, running in $\tilde
O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$
notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
  As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
$(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3
+ \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde
O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time.
  The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list
coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or
$\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of
arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$
rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly.
