Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

Franco, John; Paull, Marvin C.

doi:10.1016/0166-218x(83)90017-3

Cited by 186 publications

(93 citation statements)

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“…However, Franco and Paull [45] pointed out that large sets of random (n, m, p)-CNF expressions, for fixed 0 < p < 1/2, are dominated by easily satisfiable expressions: that is, a random assignment of values to the variables of a random expression is a model for that expression with high probability. This result is refined somewhat in [41] where it is shown that a random assignment is a model for a random (n, m, p)-CNF expression with high probability if p > ln(m)/n and a random (n, m, p)-CNF expression is unsatisfiable with high probability if p < ln(m)/2n.…”

Section: A Digest Of Probabilistic Resultsmentioning

confidence: 99%

“…Franco and Paull in [45] (see [82] for corrections) also considered the probabilistic performance of Goldberg's variant of DPLL for random (n, m, k)-CNF expressions. They showed that for all k ≥ 3 and every fixed m/n > 0, with probability 1 − o(1), the variant takes an exponential number of steps to report a result: that is, either to report all ("cylinders" of) models, or that no model exists.…”

Section: A Digest Of Probabilistic Resultsmentioning

confidence: 99%

“…Because of this, we focus attention on that generator in this chapter. This generator (from [45]) has three parameters, n, m, and k. An expression returned by this generator contains m clauses generated uniformly, independently, and with replacement from the set of all elementary disjunctions of k literals on a set of n Boolean variables and their complements. Such an expression is referred to as a random (n, m, k)-CNF expression.…”

Section: Cnf Expressionsmentioning

confidence: 99%

See 2 more Smart Citations

Probabilistic Analysis of Satisfiability Algorithms

Franco¹,

Crama²,

Hammer³

2010

Boolean Models and Methods in Mathematics, Computer Science, and Engineering

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Section: A Digest Of Probabilistic Resultsmentioning

confidence: 99%

Section: A Digest Of Probabilistic Resultsmentioning

confidence: 99%

Section: Cnf Expressionsmentioning

confidence: 99%

See 1 more Smart Citation

Probabilistic Analysis of Satisfiability Algorithms

Franco¹,

Crama²,

Hammer³

2010

Boolean Models and Methods in Mathematics, Computer Science, and Engineering

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“…Already for k ≥ 4 this improves all previously known lower bounds for r k . In Table 1, we compare our lower bound with the best known algorithmic lower bound [15,18] and the best known upper bound [10,9,19] for some small values of k. [13], in the early 1980s, observed that r * k ≤ 2 k log 2. To see this, fix any truth assignment and observe that a random k-clause is satisfied by it with probability 1 − 2 −k .…”

Section: Theoremmentioning

confidence: 99%

“…We also thank the referees for useful comments. Part of this work was done while the authors participated in the focused research group on discrete probability at BIRS, July [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] 2003.…”

Section: Acknowledgmentsmentioning

confidence: 99%

The threshold for random 𝑘-SAT is 2^{𝑘}log2-𝑂(𝑘)

Achlioptas

Peres

2004

J. Amer. Math. Soc.

138

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Let F k ( n , m ) F_k(n,m) be a random k k -CNF formula formed by selecting uniformly and independently m m out of all possible k k -clauses on n n variables. It is well known that if r ≥ 2 k log ⁡ 2 r \geq 2^k \log 2 , then F k ( n , r n ) F_k(n,rn) is unsatisfiable with probability that tends to 1 as n → ∞ n \to \infty . We prove that if r ≤ 2 k log ⁡ 2 − t k r \leq 2^k \log 2 - t_k , where t k = O ( k ) t_k = O(k) , then F k ( n , r n ) F_k(n,rn) is satisfiable with probability that tends to 1 as n → ∞ n \to \infty . Our technique, in fact, yields an explicit lower bound for the random k k -SAT threshold for every k k . For k ≥ 4 k \geq 4 our bounds improve all previously known such bounds.

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Approximating the unsatisfiability threshold of random formulas

Kirousis

Kranakis

Kriz̧anc

et al. 1998

Random Struct. Alg.

126

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Let be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number such that if the ratio of the number of clauses over the number of variables of strictly exceeds , then is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that F 5.191. This upper bound was improved by Kamath et al. to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of is around 4.2. In this work, we define, in terms of the random formula , a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q .

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Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

Cited by 186 publications

References 1 publication

Probabilistic Analysis of Satisfiability Algorithms

Probabilistic Analysis of Satisfiability Algorithms

The threshold for random 𝑘-SAT is 2^{𝑘}log2-𝑂(𝑘)

Approximating the unsatisfiability threshold of random formulas

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