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

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Citations: | 71 - 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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### 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.

### Citations

1891 | Random Graphs - Bollobás - 1985 |

1107 |
A computing procedure for quantification theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ... in [3] proved a2 �(n) lower bound for the running time for the DPLL (for Davis, Putnam, Logemann,s446 KAPORIS, KIROUSIS, AND LALAS and Loveland) procedures GUC, UC, and ORDERED-DLL; see [15, 16] and =-=[21, 22]-=-. Informally, a DPLL procedure spits a formula into two sub-formulas by setting a variable to a fixed value and recursively invokes itself on each sub-formula. For the particular case k = 3, upper bou... |

747 | Random graphs - Janson, Łuczak, et al. - 2000 |

705 |
Approximation algorithms for combinatorial problems
- Johnson
- 1973
(Show Context)
Citation Context ... heuristic [13] was analyzed. According to [5], vertices of maximum degree are given priority, but only in case they can be legally colored by 2 of 3 possible colors. Also in [36] Johnson’s heuristic =-=[43]-=- is evaluated experimentally. This heuristic selects at each free step both a literal τ and its negation τ on the basis of their corresponding degrees among 3,2-clauses. Algorithm Greedy is a simplifi... |

557 |
A machine program for theorem-proving
- Davis, Logemann, et al.
- 1962
(Show Context)
Citation Context ... in [3] proved a2 �(n) lower bound for the running time for the DPLL (for Davis, Putnam, Logemann,s446 KAPORIS, KIROUSIS, AND LALAS and Loveland) procedures GUC, UC, and ORDERED-DLL; see [15, 16] and =-=[21, 22]-=-. Informally, a DPLL procedure spits a formula into two sub-formulas by setting a variable to a fixed value and recursively invokes itself on each sub-formula. For the particular case k = 3, upper bou... |

363 |
New methods to color the vertices of a graph
- Brélaz
- 1979
(Show Context)
Citation Context ... 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 by 2 of 3 possible colors. Also in [36] Johnson’s heuristic [43] is evaluat... |

224 | and easy distributions of sat problems
- Mitchell, Selman, et al.
- 1992
(Show Context)
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... |

215 |
Many hard examples for resolution
- Chvátal, Szemerédi
- 1988
(Show Context)
Citation Context ...erns the complexity to compute a satisfying assignment, or on the contrary, to prove that none exists near the conjectured threshold value. To this end, Haken, Urquhardt, and Chvátal and Szemerédi in =-=[18, 39, 66]-=- were led to the conclusion that for k-CNF formulas of density rk > 2 k ln 2 any resolution proof of unsatisfiability contains at least (1 + ɛ) n clauses. Monasson et al, using statistical mechanics, ... |

183 |
Determining computational complexity from characteristic ‘phase transitions
- Monasson, Zecchina, et al.
- 1999
(Show Context)
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... |

171 | The asymptotic number of labeled graphs with given degree sequences - Bender, Canfield - 1978 |

164 | thresholds of graph properties, and the k-sat problem
- Sharp
- 1999
(Show Context)
Citation Context ...jectured that for each k ≥ 2 there exists a critical clauses-to-variables ratio r∗ k such that almost all k-CNF formulas φn,⌊rn⌋ with ratio (r > r∗ k )r < r∗ k are (un)-satisfiable, as n →∞. Friedgut =-=[32]-=- proved that for each k ≥ 2 there exists a sequence of threshold values r∗ k (n), depending on the number n of variables, such that for any ɛ>0 almost all k-CNF formulas (φn,⌊(r∗ k (n)+ɛ)n⌋)φn,⌊(r∗ k ... |

163 |
Mick gets some (the odds are on his side
- Chvátal, Reed
- 1992
(Show Context)
Citation Context ...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 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... |

157 | Models of random regular graphs
- Wormald
- 1999
(Show Context)
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 ... |

153 | Critical behavior in the satisfiability of random boolean expressions
- Kirkpatrick, Selman
- 1994
(Show Context)
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... |

145 |
Analytic and algorithmic solution of random satisfiability problems
- Mézard, Parisi, et al.
- 2002
(Show Context)
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, ... |

137 | Analysis of two simple heuristics on a random instance of k-SAT
- Frieze, Suen
- 1990
(Show Context)
Citation Context ... 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 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 denot... |

120 |
Differential equations for random processes and random graphs, Annals of Applied Probability 5
- Wormald
- 1995
(Show Context)
Citation Context ...e write down the system of (h+3) differential equations, the solution of which approximates within o(1) the dynamics of the algorithm per round in subSection 4I. We employ a theorem proved by Wormald =-=[72]-=-, which helps us to approximate the dynamics of the algorithm with the solution of the system of differential equations, with high probability; see sub-Section 4H. We implement the d.e. until we reach... |

107 | Sudden emergence of a giant k-core in a random graph
- Pittel, Spencer, et al.
- 1996
(Show Context)
Citation Context ...in the formula (irrespective of the clause sizes) while τ is random.sPROBABILISTIC ANALYSIS OF A SATISFIABILITY ALGORITHM 449 Finally, we were motivated to put together all heavy literals from [14], (=-=[64]-=-), where pure literals (light vertices) are set to True (deleted) in order to find a satisfying truth assignment of a random formula (the k core of a graph), respectively. B. Rounds The algorithm proc... |

103 |
A threshold for unsatisfiability
- Goerdt
- 1996
(Show Context)
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 ... |

100 | Tail bounds for occupancy and the satisfiability threshold conjecture. Randomized Structure and Algorithms
- Kamath, Motwani, et al.
- 1995
(Show Context)
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 ... |

99 | Average-case analysis of algorithms and data structures - Flajolet, Vitter - 1990 |

92 | Probabilistic analysis of a generalization of the unit-clause literal selection heuristics
- Chao, Franco
- 1990
(Show Context)
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... |

89 | Typical random 3-SAT formulae and the satisfiability threshold
- Dubois, Boufkhad, et al.
- 2000
(Show Context)
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 ... |

88 |
On the satisfiability and maximum satisfiability of random 3-CNF formulas
- Broder, Frieze, et al.
- 1993
(Show Context)
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... |

81 | Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms
- Kirousis, Kranakis, et al.
- 1998
(Show Context)
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 ... |

75 |
The random K-satisfiability problem: from an analytic solution to an efficient algorithm cond-mat 0207194
- Mézard, Zecchina
(Show Context)
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, ... |

75 |
examples for resolution
- Urquhart, Hard
- 1987
(Show Context)
Citation Context ...erns the complexity to compute a satisfying assignment, or on the contrary, to prove that none exists near the conjectured threshold value. To this end, Haken, Urquhardt, and Chvátal and Szemerédi in =-=[18, 39, 66]-=- were led to the conclusion that for k-CNF formulas of density rk > 2 k ln 2 any resolution proof of unsatisfiability contains at least (1 + ɛ) n clauses. Monasson et al, using statistical mechanics, ... |

70 | A Survey of Lower Bounds for Random 3-SAT via Differential Equations
- Achlioptas
(Show Context)
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]... |

67 | Optimal myopic algorithms for random 3-SAT
- Achlioptas, Sorkin
- 2000
(Show Context)
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... |

66 | Probabilistic analysis of two heuristics for the 3-satisfiability problem - Chao, Franco - 1986 |

65 |
Probabilistic analysis of the Davis-Putnam procedure for solving the satisfiability problem, Discrete Applied Mathematics 5:77–87
- Franco, Paul
- 1983
(Show Context)
Citation Context ...n→∞r ∗ k (n) = inf{rk :Pr[φn,⌊r k n⌋is satisfiable → 0]}. ≤ r∗+ k ,ifr ∗ k exists. 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 fa... |

62 | The SAT Phase Transition
- Gent, Walsh
- 1994
(Show Context)
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... |

55 |
A general upper bound for the satisfiability threshold of random r-sat formulae
- Dubois, Boufkhad
- 1997
(Show Context)
Citation Context |

54 |
D.B.: The scaling window of the 2-SAT transition. Random Structures and Algorithms
- Bollobas, Borgs, et al.
- 2001
(Show Context)
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 ... |

52 | 2+p-sat: Relation of typical-case complexity to the nature of the phase transition, Random Structures and Algorithms 3
- Monasson, Zecchina, et al.
- 1999
(Show Context)
Citation Context |

51 | A sharp threshold in proof complexity - Achlioptas, Beame, et al. - 2001 |

49 |
Stable marriage and its relation to other combinatorial problems
- Knuth
- 1976
(Show Context)
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... |

49 | Generating Random Regular Graphs Quickly
- Steger, Wormald
- 1999
(Show Context)
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 ... |

44 |
Branching processes and their applications in the analysis of tree structures and tree algorithms
- Devroye
- 1998
(Show Context)
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... |

42 | The asymptotic order of the random k-sat threshold
- Achlioptas, Moore
(Show Context)
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... |

42 | Bounding the unsatisfiability threshold of random 3-SAT
- Janson, Stamatiou, et al.
- 2000
(Show Context)
Citation Context |

39 | Some problems in the enumeration of labelled graphs - Wormald - 1978 |

32 | Analysis of a List-Coloring Algorithm on a Random Graph - Achlioptas, Molloy - 1997 |

28 |
Experimental results on the crossover point
- Crawford, Auton
- 1996
(Show Context)
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... |

26 |
Setting two variables at a time yields a new lower bound for random 3-SAT
- Achlioptas
- 2000
(Show Context)
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 | Random k-SAT: Two moments suffice to cross a sharp threshold, preprint
- Achlioptas, Moore
- 2002
(Show Context)
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... |

25 | Di erential equations for random processes and random - Wormald - 1995 |

22 | Random k-Sat: A tight threshold for moderately growing k - Frieze, Wormald - 2005 |

21 | Almost all graphs with average degree 4 are 3-colorable. STOC
- Achlioptas, Moore
(Show Context)
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... |

21 | Results related to thresholds phenomena research in satisfiability: lower bounds
- Franco
- 2001
(Show Context)
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]... |