Distributed Coloring of Hypergraphs

For any integer r≥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 poly(log⁡ log⁡ n)-round randomized LOCAL algorithm that computes a {\displaystyle {\mathcal O(Δ1/(r−1))-coloring w.h.p. This is tight up to polynomial factors of the time complexity as Ω(logΔ⁡log n) distributed rounds are necessary for even obtaining a Δ-coloring, where Δ is the maximum degree, and tight up to logarithmic factors of the number of colors, as Θ((Δ/log⁡Δ)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.