The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits
For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. Motivated by research on heuristics and the satisfiability threshold, in 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs . They found dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Their results were refined by Makino et al. . Recently, we were able to establish the trichotomy .
Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the connectivity problems: on one side, the diameter is linear and both problems are in P, while on the other, the diameter can be exponential and the problems are PSPACE-complete.
KeywordsBoolean Function Constraint Satisfaction Problem Propositional Formula Solution Graph Boolean Circuit
Unable to display preview. Download preview PDF.
- 1.Böhler, E., Creignou, N., Reith, S., Vollmer, H.: Playing with boolean blocks, part i: Posts lattice with applications to complexity theory. In: SIGACT News (2003)Google Scholar
- 2.Fu, Z., Malik, S.: Extracting logic circuit structure from conjunctive normal form descriptions. In: 20th International Conference on VLSI Design, Held Jointly with 6th International Conference on Embedded Systems, pp. 37–42. IEEE (2007)Google Scholar
- 11.Post, E.L.: The Two-Valued Iterative Systems of Mathematical Logic(AM-5), vol. 5. Princeton University Press (1941)Google Scholar
- 12.Reith, S., Wagner, K.W.: The complexity of problems defined by Boolean circuits (2000)Google Scholar
- 13.Schaefer, T.J.: The complexity of satisfiability problems. In: STOC 1978, pp. 216–226 (1978)Google Scholar
- 14.Schnoor, H.: Algebraic techniques for satisfiability problems. Ph.D. thesis, Universität Hannover (2007)Google Scholar
- 15.Schwerdtfeger, K.W.: A computational trichotomy for connectivity of boolean satisfiability. ArXiv CoRR abs/1312.4524 (2013), extended version of a paper submitted to the JSAT Journal, http://arxiv.org/abs/1312.4524
- 16.Schwerdtfeger, K.W.: The connectivity of boolean satisfiability: Dichotomies for formulas and circuits. ArXiv CoRR abs/1312.6679 (2013), extended version of this paper, http://arxiv.org/abs/1312.6679
- 17.Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer-Verlag New York, Inc. (1999)Google Scholar
- 18.Wu, C.A., Lin, T.H., Lee, C.C., Huang, C.Y.R.: Qutesat: a robust circuit-based sat solver for complex circuit structure. In: Proceedings of the Conference on Design, Automation and Test in Europe, EDA Consortium, pp. 1313–1318 (2007)Google Scholar