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 O~(rlogΔ+log2Δ+log∗n) rounds. (Here, the O~()
notations hides polyloglog Δ and 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+ϵ) approximation algorithm using O~(logΔ/ϵ3+polylog(1/ϵ,loglogn)) randomized time and O~(log2Δ/ϵ4+log∗n/ϵ) 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 (2Δ−1)-edge-list
coloring in O~(log2Δlogn) rounds deterministically or
O~((loglogn)3) rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most ⌈(1+ϵ)λ⌉ for a graph of
arboricity λ; for fixed ϵ this runs in O~(log6n)
rounds deterministically or O~(log3n) rounds randomly