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We give new algorithms based on Markov chains to sample and approximatelycount satisfying assignments to $k$-uniform CNF formulas where each variableappears at most $d$ times. For any $k$ and $d$ satisfying $kd<n^{o(1)}$ and$k\ge 20\log k + 20\log d + 60$, the new sampling algorithm runs in close tolinear time, and the counting algorithm runs in close to quadratic time.  Our approach is inspired by Moitra (JACM, 2019) which remarkably utilizes theLov\'{a}sz local lemma in approximate counting. Our main technical contributionis to use the local lemma to bypass the connectivity barrier in traditionalMarkov chain approaches, which makes the well developed MCMC method applicableon disconnected state spaces such as SAT solutions. The benefit of our approachis to avoid the enumeration of local structures and obtain fixed polynomialrunning times, even if $k=\omega(1)$ or $d=\omega(1)$.

Fast sampling and counting k-SAT solutions in the local lemma regime