Streaming Hypergraph Coloring via Palette Sparsification

Abstract

For every fixed $k\ge2$, we give a randomized one-pass insertion-only algorithm that colors an $n$-vertex $k$-uniform hypergraph of maximum degree $Δ$ with $O(Δ^{1/(k-1)})$ colors using $\widetilde O_k(n)$ bits of working memory. As a graph-theoretic result of independent interest, we also prove a tight palette-sparsification theorem for general uniform hypergraphs. Independently sampled lists of $Θ(\sqrt{\log n})$ colors from a palette of size $O(Δ^{1/(k-1)})$ preserve colorability with high probability; the list-size dependence is asymptotically optimal. These results extend to bounded-rank hypergraphs. We complement the algorithm with a deterministic lower bound: for every fixed polylogarithmic semi-streaming space bound, there are polylogarithmic values of $Δ$ for which any deterministic one-pass algorithm requires $\exp(Δ^{Ω(1)})$ colors.