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

arXiv: https://arxiv.org/abs/1911.01319 · HTML: https://mlsanity.com/paper/1911.01319/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2019.

## Built on by (most cited first, 5 of 5)

- [Softened Symbol Grounding for Neuro-symbolic Systems](https://mlsanity.com/paper/2403.00323/) (2024)
- [Learning Hard-Constrained Models with One Sample](https://mlsanity.com/paper/2311.03332/) (2023)
- [Learning CNF formulas from uniform random solutions in the local lemma regime](https://mlsanity.com/paper/2511.02487/) (2025)
- [One-Shot Learning for k-SAT](https://mlsanity.com/paper/2502.07135/) (2025)
- [Learning CNF Formulas from Uniform Random Solutions: Near-Tight Sample Complexity for Valiant's Algorithm](https://mlsanity.com/paper/2609.15268/) (2026)

## Abstract (arXiv)

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k-uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd<no(1) and k≥ 20log k + 20log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019) which remarkably utilizes the Lov'asz local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k=ω(1) or d=ω(1).