Sampling Lovász Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time
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We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. The expected running time of our algorithm is near-linear in n and a fixed polynomial in Δ, where n is the number of variables and Δ is the max degree of constraints. Previously, up to similar conditions, sampling algorithms with running time polynomial in both n and Δ, only existed for the almost atomic case, where each constraint is violated by a small number of forbidden local configurations. Our sampling approach departs from all previous fast algorithms for sampling LLL, which were based on Markov chains. A crucial step of our algorithm is a recursive marginal sampler that is of independent interests. Within a local lemma regime, this marginal sampler can draw a random value for a variable according to its marginal distribution, at a local cost independent of the size of the CSP.
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