Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known many industrial instances, even with a great number variables, can be solved by clever solver in reasonable amount time. This not possible, general, classical randomly generated provide different model called \emph{scale-free instances}. based use non-uniform probability distribution $P(i)\sim i^{-\beta}$ select variable $i$, where $\beta$ parameter model. results into formulas occurrences $k$ variables follows power-law $P(k)\sim k^{-\delta}$ $\delta = 1 + 1/\beta$. property has been observed most For $\beta=0$, our extends prove existence SAT-UNSAT phase transition phenomenon for scale-free 2-SAT $\beta<1/2$ when clause/variable ratio $m/n=\frac{1-2\beta}{(1-\beta)^2}$. also k-SAT are unsatisfiable high clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies SAT/UNSAT phenomena vanishes $\beta>1-1/k$, and due small core clauses. The proof this result suggests that, unsatisfiability may cores Finally, we show how will allow us generate interest testing purposes.