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.

[1]  Naftali Tishby,et al.  The Statistical Mechanics of k-Satisfaction , 1993, NIPS.

[2]  Yannis C. Stamatiou,et al.  Random Constraint Satisfaction a More Accurate Picture , 2022 .

[3]  Jun Gu,et al.  Global Optimization for Satisfiability (SAT) Problem , 1994, IEEE Trans. Knowl. Data Eng..

[4]  Toby Walsh,et al.  Random Constraint Satisfaction: Flaws and Structure , 2004, Constraints.

[5]  James M. Crawford,et al.  Experimental Results on the Crossover Point inSatis ability , 1993 .

[6]  Andrew J. Parkes,et al.  Clustering at the Phase Transition , 1997, AAAI/IAAI.

[7]  Bart Selman,et al.  Balance and Filtering in Structured Satisfiable Problems , 2001, IJCAI.

[8]  Toby Walsh,et al.  Search in a Small World , 1999, IJCAI.

[9]  Bart Selman,et al.  Problem Structure in the Presence of Perturbations , 1997, AAAI/IAAI.

[10]  Berthe Y. Choueiry,et al.  How the Level of Interchangeability Embedded in a Finite Constraint Satisfaction Problem Affects the Performance of Search , 2001, Australian Joint Conference on Artificial Intelligence.

[11]  Wayne J. Pullan,et al.  Towards Fewer Parameters for SAT Clause Weighting Algorithms , 2002, Australian Joint Conference on Artificial Intelligence.

[12]  Igor L. Markov,et al.  Solving difficult SAT instances in the presence of symmetry , 2002, Proceedings 2002 Design Automation Conference (IEEE Cat. No.02CH37324).

[13]  Michael Molloy,et al.  Models for Random Constraint Satisfaction Problems , 2003, SIAM J. Comput..

[14]  Tad Hogg,et al.  Which Search Problems Are Random? , 1998, AAAI/IAAI.

[15]  Toby Walsh,et al.  Search on High Degree Graphs , 2001, IJCAI.

[16]  J. Schmee.  Applied Statistics—A Handbook of Techniques , 1984 .

[17]  David G. Mitchell,et al.  Finding hard instances of the satisfiability problem: A survey , 1996, Satisfiability Problem: Theory and Applications.