## Abstract

Let (C_{1},C′_{1}),(C _{2}, C′_{2}),...,(C _{m} ,C′_{m} ) be a sequence of ordered pairs of 2CNF clauses chosen uniformly at random (with replacement) from the set of all 4(_{2}^{n} clauses on n variables. Choosing exactly one clause from each pair defines a probability distribution over 2CNF formulas. The choice at each step must be made on-line, without backtracking, but may depend on the clauses chosen previously. We show that there exists an on-line choice algorithm in the above process which results in a satisfiable 2CNF formula as long as m/n ≤ (1000/999)^{1/4}. This contrasts with the well-known fact that a random m-clause formula constructed without the choice of two clauses at each step is unsatisfiable whenever m/n > 1. Thus the choice algorithm is able to delay satisfiability of a random 2CNF formula beyond the classical satisfiability threshold. Choice processes of this kind in random structures are known as "Achlioptas processes." This paper joins a series of previous results studying Achlioptas processes in different settings, such as delaying the appearance of a giant component or a Hamilton cycle in a random graph. In addition to the on-line setting above, we also consider an off-line version in which all m clause-pairs are presented in advance, and the algorithm chooses one clause from each pair with knowledge of all pairs. For the off-line setting, we show that the two-choice satisfiability threshold for k-SAT for any fixed k coincides with the standard satisfiability threshold for random 2k-SAT.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 13th International Workshop, APPROX 2010 and 14th International Workshop, RANDOM 2010, Proceedings |

Pages | 710-723 |

Number of pages | 14 |

DOIs | |

State | Published - 15 Nov 2010 |

Externally published | Yes |

Event | 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010 - Barcelona, Spain Duration: 1 Sep 2010 → 3 Sep 2010 |

### 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 | 6302 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2010 and 14th International Workshop on Randomization and Computation, RANDOM 2010 |
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Country/Territory | Spain |

City | Barcelona |

Period | 1/09/10 → 3/09/10 |