Towards the sampling Lov\'asz Local Lemma

Let $\Phi = (V, \mathcal{C})$ be a constraint satisfaction problem on variables $v_1,\dots, v_n$ such that each constraint depends on at most $k$ variables and such that each variable assumes values in an alphabet of size at most $[q]$. Suppose that each constraint shares variables with at most $\Delta$ constraints and that each constraint is violated with probability at most $p$ (under the product measure on its variables). We show that for $k, q = O(1)$, there is a deterministic, polynomial time algorithm to approximately count the number of satisfying assignments and a randomized, polynomial time algorithm to sample from approximately the uniform distribution on satisfying assignments, provided that $$C\cdot q^{3}\cdot k \cdot p \cdot \Delta^{7} < 1, \quad \text{where }C \text{ is an absolute constant.}$$ Previously, a result of this form was known essentially only in the special case when each constraint is violated by exactly one assignment to its variables. For the special case of $k$-CNF formulas, the term $\Delta^{7}$ improves the previously best known $\Delta^{60}$ for deterministic algorithms [Moitra, J.ACM, 2019] and $\Delta^{13}$ for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly $q$-coloring $k$-uniform hypergraphs, the term $\Delta^{7}$ improves the previously best known $\Delta^{14}$ for deterministic algorithms [Guo et al., SICOMP, 2019] and $\Delta^{9}$ for randomized algorithms [Feng et al., arXiv, 2020].