Distributed approximation algorithms for maximum matching in graphs and hypergraphs
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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 Õ( r Δ) for the randomized algorithm and Õ(r Δ + ^3 Δ) 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+ϵ approximation algorithm running in Õ(Δ) randomized time and Õ(^3 Δ + ^* n) deterministic time. These are significantly faster than prior 1+ϵ-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 (1+ϵ) a ; this runs in Õ(^7 n ^3 a) rounds deterministically or Õ(^3 n ) rounds randomly.
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