A sampling Lovász Local Lemma

We give an approximately uniform sampler for satisfying assignments of constraint satisfaction problems that satisfy $4\mathrm{e}p(\mathrm{\Delta }+1{)}^2\le 1$, where $p$ is the largest constraint-violation probability under the uniform product distribution, and $\mathrm{\Delta }$ is the maximum degree of the dependency graph. The algorithm invokes the recent efficient approximate counting algorithm of Liu, Wang, Yin, Zhang, and Zhou as a subroutine and returns a satisfying assignment sampled within total-variation distance $\epsilon$ of the uniform distribution in $(n+m\mathrm{/}\epsilon {)}^{O(k\mathrm{\Delta }\log D)}$ time, where $n$ and $m$ are the numbers of variables and constraints, $D$ is the common domain size, and $k$ bounds the constraint arity.