Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

Abstract: 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 $$\tilde O(r \log \Delta + \log² \Delta + \log^* n)$$ 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 $$\tilde O(\log \Delta / \epsilon³ + \text{polylog}(1/\epsilon, \log \log n))$$ randomized time and $$\tilde O(\log² \Delta / \epsilon⁴ + \log^*n / \epsilon)$$ 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 $\tilde O(\log² \Delta \log n)$ 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 $\tilde O(\log⁶ n)$ rounds deterministically or $\tilde O(\log³ n )$ rounds randomly.