## The Probabilistic Analysis of a Greedy Satisfiability Algorithm (2002)

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Citations: | 70 - 5 self |

### BibTeX

@MISC{Kaporis02theprobabilistic,

author = {Alexis C. Kaporis and Lefteris M. Kirousis and Efthimios G. Lalas},

title = {The Probabilistic Analysis of a Greedy Satisfiability Algorithm},

year = {2002}

}

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### OpenURL

### Abstract

Consider the following simple, greedy Davis-Putnam algorithm applied to a random 3CNF formula of fixed density (clauses to variables ratio): Arbitrarily select and set to True a literal that appears in as many clauses as possible, irrespective of their size (and irrespective of the number of occurrences of the negation of the literal). Delete these clauses from the formula, and also delete the negation of this literal from any clauses it appears. Repeat. If however unit clauses ever appear, then first repeatedly and in any order set the literals in them to True and delete and shrink clauses accordingly, until no unit clause remains. Also if at any step an empty clause appears, then do not backtrack, but just terminate the algorithm and report failure. A slight modification of this algorithm is probabilistically analyzed in this paper (rigorously). It is proved that for random formulas of n variables and density up to 3.42, it succeeds in producing a satisfying truth assignment with bounded away from zero probability, as n approaches infinity. Therefore the satisfiability threshold is at least 3.42.

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Citation Context ...sts. Franco and Paull pioneered the study of random k-CNF formulas and proved the general upper bound r ∗+ k < 2k ln 2 in [31]. Then Chao and Franco established the general lower �k−22k ∗− /k < rk in =-=[16]-=-. These results suggested the simple law r ∗− k = r∗ k = r ∗+ k ∼ 2k ln 2. A series of experimental results comes up in favor of the threshold conjecture, see [20, 25, 58]. Monasson and Zecchina, usin... |

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Citation Context ...y setting a variable to a fixed value and recursively invokes itself on each sub-formula. For the particular case k = 3, upper bounds to r ∗+ 3 have been proven using probabilistic counting arguments =-=[26, 27, 42, 44, 47, 50, 53]-=-; see the surveys [24, 51] about the techniques employed. Dubois, Boufkhad, and Mandler proved r ∗+ 3 < 4.506 as the current best upper bound in [27]. As a corollary of [32], to prove that c < r ∗− 3 ... |

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Citation Context ...y setting a variable to a fixed value and recursively invokes itself on each sub-formula. For the particular case k = 3, upper bounds to r ∗+ 3 have been proven using probabilistic counting arguments =-=[26, 27, 42, 44, 47, 50, 53]-=-; see the surveys [24, 51] about the techniques employed. Dubois, Boufkhad, and Mandler proved r ∗+ 3 < 4.506 as the current best upper bound in [27]. As a corollary of [32], to prove that c < r ∗− 3 ... |

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Citation Context ...ition correlates to the running time until a satisfying truth assignment is returned, by heuristics that are based on the Davis–Putnam simplification rule [62, 63]. Furthermore, Mézard and colleagues =-=[55, 56]-=- suggest a linear time algorithmic criterion that may improve the lower bound on r ∗− k . Achlioptas, Beame, and Molloy in [3] proved a2 �(n) lower bound for the running time for the DPLL (for Davis, ... |

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Citation Context ...th assignment with the probability of at least a positive constant. In this vein, Davis–Putnam algorithms of increasing sophistication were rigorously analyzed [1, 9, 14, 15, 17, 33]; see the surveys =-=[2, 30]-=- describing in detail various heuristics and techniques on their analysis. The best previous lower bound for the satisfiability threshold thus obtained is 3.26 < r ∗− 3 by Achlioptas and Sorkin in [9]... |

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Citation Context ...ormula of density c has a satisfying truth assignment with the probability of at least a positive constant. In this vein, Davis–Putnam algorithms of increasing sophistication were rigorously analyzed =-=[1, 9, 14, 15, 17, 33]-=-; see the surveys [2, 30] describing in detail various heuristics and techniques on their analysis. The best previous lower bound for the satisfiability threshold thus obtained is 3.26 < r ∗− 3 by Ach... |

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Citation Context ...nd for general k-CNF formulas in [33]. Wilson in [70] proved that for each k ≥ 2 the characteristic width of the phase transition is at least �(n 1/2 ), contradicting a number of empirical results in =-=[35, 48, 49, 62, 63]-=-. The width denotes the amount of extra clauses needed to be added in the random formula for the probability of satisfiability to drop from 1 − ɛ to ɛ. In a recent advance, Frieze and Wormald [34] pro... |

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Citation Context |

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Citation Context ...ica method from statistical mechanics, predicted r ∗ k ∼ 2k ln 2 in [61]. Chvátal and Reed in [17] proved the simple law 1 4 2k /k < r ∗− k and further proved that r ∗− 2 = r∗ 2 = r∗+ 2 = 1; also see =-=[12, 17, 37, 67, 68]-=-. Frieze and Suen improved to zk2 k /k < r ∗− k , where zk = O(1) depending on k, as the best algorithmic lower bound for general k-CNF formulas in [33]. Wilson in [70] proved that for each k ≥ 2 the ... |

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Citation Context ...refined in [12, 73]. The problem of handling degree information of a random structure has attracted a lot of interest lately. Of particular interest is the issue of generating random r-regular graphs =-=[71, 74]-=-. In such a graph all n vertices have degree r and is constructed by creating r copies of each of the n vertices (or hanging semi-edges) and choosing a random matching on these semi-edges. As long as ... |

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Citation Context ...ability to drop from 1 − ɛ to ɛ. In a recent advance, Frieze and Wormald [34] proved that 2 k ln 2 < r ∗− k as k − log 2 n → ∞, employing a second moment argument. Independently, Achlioptas and Moore =-=[6]-=- also applied the second moment method to prove that 2k 2 ln 2 − zk < r ∗− k for any fixed value of k ≥ 2, where zk > 0 is constant and depends on k. Recently, Achlioptas and Peres refined this method... |

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Citation Context ...onstant during a round (we will elaborate on this point below). Then the generation of the 1-clauses during the forced steps of the round follows the pattern of a Galton–Watson branching process (see =-=[23]-=-). Such a process starts with a pater familias or root (or alma mater) and then at every step all individuals born at the previous step generate a number of offspring. The number of offspring in a Gal... |

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Citation Context ... be specified uniformly at random among the 3|C3|+2|C2| possible literal occurrences. This is done in an analogous manner as the content of a card is revealed in the expository card game presented in =-=[52]-=-. Using Model B, we can Select & Set to True a random literal (or a random literal occurrence in a clause of specified length), amounting to model’s B first (or second) kind of permissible atomic step... |

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Citation Context ...ormula of density c has a satisfying truth assignment with the probability of at least a positive constant. In this vein, Davis–Putnam algorithms of increasing sophistication were rigorously analyzed =-=[1, 9, 14, 15, 17, 33]-=-; see the surveys [2, 30] describing in detail various heuristics and techniques on their analysis. The best previous lower bound for the satisfiability threshold thus obtained is 3.26 < r ∗− 3 by Ach... |

26 |
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Citation Context ...e general lower �k−22k ∗− /k < rk in [16]. These results suggested the simple law r ∗− k = r∗ k = r ∗+ k ∼ 2k ln 2. A series of experimental results comes up in favor of the threshold conjecture, see =-=[20, 25, 58]-=-. Monasson and Zecchina, using the non-rigorous replica method from statistical mechanics, predicted r ∗ k ∼ 2k ln 2 in [61]. Chvátal and Reed in [17] proved the simple law 1 4 2k /k < r ∗− k and furt... |

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Citation Context ...ify the probabilistic analysis, many papers have adopted slight modifications of this model, which may allow repeated or complementary literals in a clause and repetitions of clauses. A popular model =-=[2, 7, 14, 27, 34, 59]-=-, not restricted to the study of algorithmic issues concerning k-SAT, is the following: We construct a random φn,m by selecting, for each of the km total clause positions in it, a literal in L uniform... |

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Citation Context ...th assignment with the probability of at least a positive constant. In this vein, Davis–Putnam algorithms of increasing sophistication were rigorously analyzed [1, 9, 14, 15, 17, 33]; see the surveys =-=[2, 30]-=- describing in detail various heuristics and techniques on their analysis. The best previous lower bound for the satisfiability threshold thus obtained is 3.26 < r ∗− 3 by Achlioptas and Sorkin in [9]... |

20 | Almost all graphs with average degree 4 are 3-colorable
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Citation Context ...ion as if we had to set to true a random literal. We were motivated to give priority to large degrees from [2, 9], where the need to capitalize on variable-degree information was pointed out and from =-=[5]-=-, where, in the context of the 3-coloring problem, the Brélaz heuristic [13] was analyzed. According to [5], vertices of maximum degree are given priority, but only in case they can be legally colored... |