Distributed Coloring of Hypergraphs.
SIROCCO(2023)
摘要
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.
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关键词
hypergraphs,coloring
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