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 $\mathrm{\Delta }$ if $k\ge 28$ and $q>357{\mathrm{\Delta }}^{\frac{14}{k-14}}$ . We also obtain a polynomial-time almost uniform sampler if $q>931{\mathrm{\Delta }}^{\frac{16}{k-16\mathrm{/}3}}$. These are the first approximate counting and sampling algorithms in the regime $q\ll \mathrm{\Delta }$ (for large $\mathrm{\Delta }$ 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 $\mathrm{\Delta }$ in Moitra’s approach.