## Abstract

We study typical case properties of the 1-in-3 satisfiability problem, the Boolean satisfaction problem, where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 satisfiability and exact 3-cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region and develop a one-step replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.

Original language | English |
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Article number | 011101 |

Journal | Physical Review E |

Volume | 76 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2 Jul 2007 |

### Bibliographical note

©2007 The American Physical Society. Phase diagram of the 1-in-3 satisfiability problemJack Raymond, Andrea Sportiello, and Lenka Zdeborová

Phys. Rev. E 76, 011101 – Published 2 July 2007