Towards the sampling Lovász Local Lemma
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Let Φ = (V, 𝒞) be a constraint satisfaction problem on variables v_1,…, 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 Δ 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· q^3· k · p ·Δ^7 < 1, where C 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 Δ^7 improves the previously best known Δ^60 for deterministic algorithms [Moitra, J.ACM, 2019] and Δ^13 for randomized algorithms [Feng et al., arXiv, 2020]. For the special case of properly q-coloring k-uniform hypergraphs, the term Δ^7 improves the previously best known Δ^14 for deterministic algorithms [Guo et al., SICOMP, 2019] and Δ^9 for randomized algorithms [Feng et al., arXiv, 2020].
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