Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs

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 $\tilde O()$ notation hides polyloglog $\Delta$ and polylog $r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris, and Kuhn [On derandomizing local distributed algorithms, in Proceedings of the $59$th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 662--673]. The first main application is to nearly optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$-approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 +$ 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, Harris, and Kuhn [On derandomizing local distributed algorithms, in Proceedings of the $59$th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 662--673] 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.