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 (O) over tilde (r log Delta + log(2) Delta + log* n) rounds. (Here, the (O) over tilde() 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 59th 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 (O) over tilde (log Delta/epsilon(3) + polylog(1/epsilon, log log n)) randomized time and (O) over tilde (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 59th 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 (O) over tilde (log(2) Delta log n) rounds deterministically or (O) over tilde( (log log n)(3)) rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most inverted rihgt perpendicular(1 + epsilon)lambda inverted left perpendicular for a graph of arboricity lambda; for fixed c this runs in (O) over tilde (log(6) n) rounds deterministically or (O) over tilde (log(3) n) rounds randomly.