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arXiv · 1711.03396

Counting hypergraph colorings in the local lemma regime

Heng Guo·Chao Liao·Pinyan Lu·Chihao Zhang
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Abstract

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of $q$-colorings for $k$-uniform hypergraphs with maximum degree $Δ$ if $k\ge 28$ and $q >357Δ^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931Δ^{\frac{16}{k-16/3}}$. These are the first approximate counting and sampling algorithms in the regime $q\llΔ$ (for large $Δ$ and $k$) without any additional assumptions. Our method is based on the recent work of Moitra (STOC, 2017). One important contribution of ours is to remove the dependency of $k$ and $Δ$ in Moitra's approach.

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Heng Guo, Chao Liao, Pinyan Lu, Chihao Zhang. 2019-05-31. Counting hypergraph colorings in the local lemma regime. https://arxiv.org/abs/1711.03396

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