Distributed Coloring of Hypergraphs

For any integer $$r \ge 2$$, a linear r-uniform hypergraph is a generalization of ordinary graphs, where edges contain r vertices and two edges intersect in at most one node. We consider the problem of coloring such hypergraphs in several constrained models of computing, i.e., computing a partition such that no edge is fully contained in the same class. In particular, we give a $$\textrm{poly}(\log \log n)$$ -round randomized Local algorithm that computes a $$O(\varDelta ^{1/(r-1)})$$ -coloring w.h.p. This is tight up to polynomial factors of the time complexity as $$\varOmega (\log _\varDelta \log n)$$ distributed rounds are necessary for even obtaining a $$\varDelta $$ -coloring, where $$\varDelta $$ is the maximum degree, and tight up to logarithmic factors of the number of colors, as $$\varTheta ((\varDelta /\log \varDelta )^{1/(r-1)})$$ colors are necessary for existence. We also give simple algorithms that run in O(1)-rounds of the Congested Clique model and in a single-pass of the semi-streaming model.