19960101
Approximating the unsatisfiability threshold of random formulas: (Extended abstract)
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Publication
Let φ be a random Boolean formula that is an instance of 3SAT. 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 φ strictly exceeds κ, then φ is almost certainly unsatisfiable. By a well known and more or less straightforward argument, it can be shown that κ < 5. 191. This upper bound was improved by Kamath, Motwani, Palem, and Spirakis to 4.758, by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of κ is around 4. 2. In this work, we define, in terms of the random formula φ, a decreasing sequence of random variables such that if the expected value of any one of them converges to zero, then φ 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 κ equal to 4. 667. From the expected value of the second term of the sequence, we get the value 4. 598. In general, by letting the expected value of further terms of this sequence converge to zero, one can, if the calculations are performed, obtain even better approximations to κ. This technique generalizes in a straightforward manner to kSAT, for k > 3.
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Persistent URL  dx.doi.org/10.1007/3540616802_44 
Citation 
Kirousis, L.M. (Lefteris M.), Kranakis, E, & Krizanc, D. (Danny). (1996). Approximating the unsatisfiability threshold of random formulas: (Extended abstract). doi:10.1007/3540616802_44
