### 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.

Original language | English (US) |
---|---|

Title of host publication | Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |

Publisher | IEEE Computer Society |

Pages | 724-731 |

Number of pages | 8 |

ISBN (Electronic) | 9781509039333 |

DOIs | |

State | Published - Dec 14 2016 |

Externally published | Yes |

Event | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 - New Brunswick, United States Duration: Oct 9 2016 → Oct 11 2016 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
---|---|

Volume | 2016-December |

ISSN (Print) | 0272-5428 |

### Other

Other | 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016 |
---|---|

Country | United States |

City | New Brunswick |

Period | 10/9/16 → 10/11/16 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

### Keywords

- Condensation transition
- Free energy
- One-step replica symmetry breaking
- Random constraint satisfaction problem
- Replica symmetry
- Satisfiability threshold

## Fingerprint Dive into the research topics of 'The Number of Solutions for Random Regular NAE-SAT'. Together they form a unique fingerprint.

## Cite this

*Proceedings - 57th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2016*(pp. 724-731). [7782987] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2016-December). IEEE Computer Society. https://doi.org/10.1109/FOCS.2016.82