We describe randomized and deterministic approximation algorithms in Linial's 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. The 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 & 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 & 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.
from cs updates on arXiv.org https://ift.tt/2O7CNdm
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