The Connectivity of Boolean Satisfiability: No-Constants and Quantified Variants

TL;DR

This paper investigates the connectivity properties of solution spaces in Boolean satisfiability problems, extending previous work to formulas without constants and with quantifiers, and classifies the complexity of related problems.

Contribution

It extends the analysis of solution graph connectivity to CNF formulas without constants and quantified variants, providing new complexity classifications.

Findings

Dichotomies for diameter and st-connectivity in new variants.

Identification of fragments with polynomial, coNP-complete, and PSPACE-complete connectivity problems.

Extension of previous dichotomy and trichotomy results to broader classes of formulas.

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, Gopalan et al. in 2006 studied connectivity properties of the solution graph and related complexity issues for constraint satisfaction problems in Schaefer's framework. 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 that we recently were able to prove. While Gopalan et al. considered CNF(S)-formulas with constants, we here look at two important variants: CNF(S)-formulas without constants, and partially quantified formulas. For the diameter and the st-connectivity question, we prove dichotomies analogous to those of Gopalan et al. in these settings. While we cannot give a complete classification for the connectivity problem yet, we identify fragments where it is in P, where it is coNP-complete, and where it is PSPACE-complete, in analogy to Gopalan et al.'s trichotomy.