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# The number of solutions for random regular NAE-SAT

## Author(s): Sly, Allan M.; Sun, Nike; Zhang, Yumeng

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 Abstract: Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems (CSPs). In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions from the statistical physics literature. Here we revisit one of these models, random regular NAE-SAT: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain “survey propagation model” associated to this problem. We believe that these methods may be applicable in a wide class of related problems. Publication Date: 2016 Citation: Sly, Allan, Sun, Nike, Zhang, Yumeng. (2016). The number of solutions for random regular NAE-SAT. 2016 IEEE 57TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS), 724 - 731. doi:10.1109/FOCS.2016.82 DOI: doi:10.1109/FOCS.2016.82 ISSN: 0272-5428 Pages: 724 - 731 Type of Material: Conference Article Journal/Proceeding Title: 2016 IEEE 57TH ANNUAL SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS) Version: Author's manuscript

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