An Investigation of Variable Relationships in 3-SAT Problems

To date, several types of structure for finite Constraint Satisfaction Problems have been investigated with the goal of either improving the performance of problem solvers or allowing efficient problem solvers to be identified. Our aim is to extend the work in this area by performing a structural analysis in terms of variable connectivity for 3-SAT problems. Initially structure is defined in terms of the compactness of variable connectivity for a problem. Using an easily calculable statistic developed to measure this compactness, a test was then created for identifying 3-SAT problems as either compact, loose or unstructured (or uniform). A problem generator was constructed for generating 3-SAT problems with varying degrees of structure. Using problems from this problem generator and existing problems from SATLIB, we investigated the effects of this type of structure on satisfiability and solvability of 3-SAT problems. For the same problem length, it is demonstrated that satisfiability and solvability are different for structured and uniform problems generated by the problem generator.