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## On the satisfiability threshold and clustering of solutions of random 3-Sat formulas (2007)

Citations: | 10 - 1 self |

### Citations

8756 |
Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference
- Pearl
- 1988
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Citation Context ...nal inputs. A detailed study of the Survey Propagation algorithm undertaken in [14] and [3] led to an interpretation of the message passing procedure as the more familiar Belief Propagation algorithm =-=[19]-=- applied to a particular probability distribution on partial assignments, i.e., assignments of values from the set {0,1, ∗}. Here ∗ is to be interpreted as “unassigned,” and a variable is allowed to b... |

195 | Analytic and algorithmic solutions to random satisfiability problems
- Mézard, Parisi, et al.
(Show Context)
Citation Context ...sign variables greedily one by one in an order that is based only on the number of positive and negative occurrences of each variable. An apparently much more powerful algorithm is Survey Propagation =-=[17, 18]-=-. In experiments on very large instances (say, with n = 10 6 variables) it finds solutions for formulas of densities only just below the conjectured threshold value α = 4.27; however, a rigorous analy... |

107 | Tail bounds for occupancy and the satisfiability threshold conjecture,” Random Structures and Algorithms 7
- Kamath, Motwani, et al.
- 1995
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Citation Context ...nd 3-clauses is satisfiable. For this sub-problem, it is natural to use one of the methods that have been previously introduced for bounding the probability of satisfiability of random 3-Sat formulas =-=[10, 5, 13, 9, 12, 6]-=-. However, the most powerful method, from [6], is very heavy numerically, and it does not seem possible to carry it out for the whole range of required densities; on the other hand, simpler methods su... |

97 | Typical random 3-SAT formulae and the satisfiability threshold - Dubois, Boufkhad, et al. - 2000 |

87 | Approximating the unsatisfiability threshold of random formulas,” Random Structures and Algorithms 12
- Kirousis, Kranakis, et al.
- 1998
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Citation Context ...nd 3-clauses is satisfiable. For this sub-problem, it is natural to use one of the methods that have been previously introduced for bounding the probability of satisfiability of random 3-Sat formulas =-=[10, 5, 13, 9, 12, 6]-=-. However, the most powerful method, from [6], is very heavy numerically, and it does not seem possible to carry it out for the whole range of required densities; on the other hand, simpler methods su... |

76 | The probabilistic analysis of a greedy satisfiability algorithm
- Kaporis, Kirousis, et al.
- 2002
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Citation Context ...satisfying assignment with high probability. It is conjectured that αc(n) is a constant (and that its value is about 4.27), but currently all that is known is that, for large n, 3.520 ≤ αc(n) ≤ 4.506 =-=[11, 8, 6]-=-. In the range of densities for which the formula is satisfiable with high probability, the interesting algorithmic question is whether we can find even one of the many satisfying assignments in polyn... |

60 | On the solution-space geometry of random constraint satisfaction problems
- Achlioptas, Ricci-Tersenghi
- 2006
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Citation Context ...euristic for random Sat. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-Sat with k ≥ 9 =-=[1]-=-. Our result implies that either this does not hold for 3-Sat or the threshold density for satisfiability in 3-Sat lies below 4.453. The main technical tool that we use is a novel simple application o... |

58 |
A general upper bound for the satisfiability threshold of random r-SAT
- Dubois, Boufkhad
- 1997
(Show Context)
Citation Context ...nd 3-clauses is satisfiable. For this sub-problem, it is natural to use one of the methods that have been previously introduced for bounding the probability of satisfiability of random 3-Sat formulas =-=[10, 5, 13, 9, 12, 6]-=-. However, the most powerful method, from [6], is very heavy numerically, and it does not seem possible to carry it out for the whole range of required densities; on the other hand, simpler methods su... |

50 |
Survey propagation as local equilibrium equations
- Braunstein, Zecchina
(Show Context)
Citation Context ...king much further ahead) in order to systematically design algorithms for more general problems with distributional inputs. A detailed study of the Survey Propagation algorithm undertaken in [14] and =-=[3]-=- led to an interpretation of the message passing procedure as the more familiar Belief Propagation algorithm [19] applied to a particular probability distribution on partial assignments, i.e., assignm... |

48 | Clustering of solutions in the random satisfiability problem - Mézard, Mora, et al. - 1972 |

44 | Bounding the unsatisfiability threshold of random 3-SAT,” Random Structures and Algorithms
- Janson, Stamatiou, et al.
- 2000
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Citation Context |

34 | The satisability threshold of random 3-SAT is at least 3.52
- Hajiaghayi, Sorkin
- 2003
(Show Context)
Citation Context ...satisfying assignment with high probability. It is conjectured that αc(n) is a constant (and that its value is about 4.27), but currently all that is known is that, for large n, 3.520 ≤ αc(n) ≤ 4.506 =-=[11, 8, 6]-=-. In the range of densities for which the formula is satisfiable with high probability, the interesting algorithmic question is whether we can find even one of the many satisfying assignments in polyn... |

23 | Selecting complementary pairs of literals - Kaporis, Kirousis, et al. - 2003 |

12 |
Random k-satisfiability: from an analytic solution to an efficient algorithm
- Mézard, Zecchina
(Show Context)
Citation Context ...sign variables greedily one by one in an order that is based only on the number of positive and negative occurrences of each variable. An apparently much more powerful algorithm is Survey Propagation =-=[17, 18]-=-. In experiments on very large instances (say, with n = 10 6 variables) it finds solutions for formulas of densities only just below the conjectured threshold value α = 4.27; however, a rigorous analy... |

9 |
Almost all graphs with 1.44 edges are 3 colourable. Random Structures and Algorithms
- Chvatal
- 1991
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Citation Context ...at there is at least one of each kind is the same as the coupon collectors probability of success, with s = ⌊an⌋ different coupons, and m ′ = (dα)n trials. We will use the following general fact from =-=[4]-=-, which was previously used in a very similar context in [12]: Let q(cN,N) denote the probability of collecting N coupons within cN trials. If c < 1, q(cN,N) = 0. Otherwise, as N goes to infinity q(cN... |

9 | The unsatisfiability threshold revisited
- Kaporis, Kirousis, et al.
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Citation Context |

6 | Coupon collectors, q-binomial coefficients and the unsatisfiability threshold - KAPORIS, KIROUSIS, et al. - 2001 |

5 |
Neccesary and sufficient conditions for sharp threhsolds of graph properties and the k-problem
- Friedgut
- 1999
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Citation Context ...of computational tasks on random inputs. In random 3-Sat the input is a formula drawn uniformly at random from all formulas of fixed density α, i.e., formulas with αn clauses on n variables. Friedgut =-=[7]-=- proved that there exists a function αc(n), known as the satisfiability threshold, such that for any positive ǫ, random formulas of density αc(n) −ǫ have satisfying assignments with high probability, ... |

2 | Pruning processes and a new characterization of convex geometries
- Ardila, Maneva
- 2007
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Citation Context ... be to use a different weight ρ for each variable, depending for example on the number of positive and negative occurrences of the variable. The corresponding generalization of Theorem 4 is proved in =-=[2]-=-. It is quite possible that the value of E[Z] in this case is significantly smaller. 5 Acknowledgments We would like to thank the anonymous referees for their detailed and very useful suggestions. Ref... |

2 |
A new perspective on survey propagation
- Maneva, Mossel, et al.
(Show Context)
Citation Context ...also (looking much further ahead) in order to systematically design algorithms for more general problems with distributional inputs. A detailed study of the Survey Propagation algorithm undertaken in =-=[14]-=- and [3] led to an interpretation of the message passing procedure as the more familiar Belief Propagation algorithm [19] applied to a particular probability distribution on partial assignments, i.e.,... |

2 | Convex geometries in k-sat problems - Ardila, Maneva - 2007 |