The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

Abstract

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 [3]. 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. [7]. Recently, we were able to establish the trichotomy [15].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.