Distributed approximation algorithms for maximum matching in graphs and hypergraphs.

We describe randomized and deterministic approximation algorithms in Linialu0027s classic LOCAL model of distributed computing to find maximum-weight matchings in hypergraphs. For a rank-$r$ hypergraph, our algorithm generates a matching within an $O(r)$ factor of the maximum weight matching. runtime is $tilde O(log r log Delta)$ for the randomized algorithm and $tilde O(r log Delta + log^3 Delta)$ for the deterministic algorithm. The randomized algorithm is a straightforward, though somewhat delicate, combination of an LP solver algorithm of Kuhn, Moscibroda Wattenhofer (2006) and randomized rounding. For the deterministic part, we extend a method of Ghaffari, Harris u0026 Kuhn (2017) to derandomize the first-moment method; this allows us to deterministically simulate an alteration-based probabilistic construction. This hypergraph matching algorithm has two main algorithmic consequences. First, we get nearly-optimal deterministic and randomized algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we obtain a $1+epsilon$ approximation algorithm running in $tilde O(log Delta)$ randomized time and $tilde O(log^3 Delta + log^* n)$ deterministic time. These are significantly faster than prior $1+epsilon$-approximation algorithms; furthermore, there are no constraints on the size of the edge weights. Second, we get an algorithm for hypergraph maximal matching, which is significantly faster than the algorithm of Ghaffari, Harris u0026 Kuhn (2017). One main consequence (along with some additional optimizations) is an algorithm which takes an arboricity-$a$ graph and generates an edge-orientation with out-degree $lceil (1+epsilon) a rceil$; this runs in $tilde O(log^7 n log^3 a)$ rounds deterministically or $tilde O(log^3 n )$ rounds randomly.