## Abstract

In this paper we show that the so-called normalized auto- correlation length of random Max r-Sat converges in probability to (1 − 1/2^{r})/r, where r is the number of literals in a clause. We also show that the correlation between the numbers of clauses satisfied by a random pair of assignments of distance d = cn, 0 ≤ c ≤ 1, converges in probability to ((1 − c)^{r} − 1/2^{r})/(1 − 1/2^{r}). The former quantity is of interest in the area of landscape analysis as a way to better understand problems and assess their hardness for local search heuristics. In [34], it has been shown that it may be calculated in polynomial time for any instance, and its mean value over all instances was discussed. Our results are based on a study of the variance of the number of clauses satisfied by a random assignment, and the covariance of the numbers of clauses satisfied by a random pair of assignments of an arbitrary distance. As part of this study, closed-form formulas for the expected value and vari- ance of the latter two quantities are provided. Note that all results are relevant to random r-Sat as well.

Original language | English |
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Title of host publication | Theory and Applications of Satisfiability Testing – SAT 2016 - 19th International Conference, Proceedings |

Editors | Daniel Le Berre, Nadia Creignou |

Publisher | Springer Verlag |

Pages | 60-76 |

Number of pages | 17 |

ISBN (Print) | 9783319409696 |

DOIs | |

State | Published - 1 Jan 2016 |

Event | 19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016 - Bordeaux, France Duration: 5 Jul 2016 → 8 Jul 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9710 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 19th International Conference on Theory and Applications of Satisfiability Testing, SAT 2016 |
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Country/Territory | France |

City | Bordeaux |

Period | 5/07/16 → 8/07/16 |

## Keywords

- Autocorrelation length
- Combinatorial
- Fitness landscapes
- Local search
- Max sat
- Optimization

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science

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