The Number of Solutions for Random Regular NAE-SAT

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....

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Bibliographic Details
Main Authors: Sly, Allan, Sun, Nike, Yumeng Zhang
Format: Conference Proceeding
Language:English
Subjects:
Analytical models
Computer science
condensation transition
Electronic mail
free energy
one-step replica symmetry breaking
Physics
Predictive models
Probability
random constraint satisfaction problem
replica symmetry
satisfiability threshold
Sun
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Summary: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.
ISSN:0272-5428
DOI:10.1109/FOCS.2016.82