Counting hypergraph colorings in the local lemma regime

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of q-colorings for k-uniform hypergraphs with maximum degree∆ if k ≥ 28 and q > 315∆ 14 k−14 . We also obtain a polynomial-time almost uniform sampler if q > 798∆ 16 k−16/3 . These are the first approximate counting and sampling algorithms in the regime q ≪ ∆ (for large ∆ 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 ∆ in Moitra’s approach.