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. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs [GKMP09], motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [Sch13].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 two connectivity problems: On one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.

A Reconfigurations Analogue of Brooks' Theorem and Its Consequences

“…As this definition suggests, for a given search problem there might be more than one way to define an edge relation of the reconfiguration graph. Reconfiguration graphs have not only been studied for coloring, but also for many other problems, including Boolean satisfiability [, , ], clique and vertex cover , independent set [, , ], list edge coloring [, ], L (2, 1)‐labeling , shortest path [, ], and subset sum ; see also a recent survey . Typical questions are as follows: is the reconfiguration graph connected; if so what is its diameter; if not what is the diameter of its (connected) components; and how difficult is it to decide whether there is a path between a pair of given solutions?…”