A Dichotomy Theorem within Schaefer for the Boolean Connectivity Problem

Gopalan et al. studied in [14] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity problems in Schaefer’s framework [26]. A set S of logical relations is Schaefer if all relations in S are either bijunctive, Horn, dual Horn, or affine. They conjectured that the connectivity problem for Schaefer is in P . We disprove their conjecture by showing that it is coNP-complete for Horn and dual Horn relations. This, together with the results in [14], implies a dichotomy theory within Schaefer and a trichotomy theory for the connectivity problem. We also show that the connectivity problem for bijunctive relations can be solved in O(min{n|φ|, T (n)}) time, where n denotes the number of variables, φ denotes the corresponding 2-CNF formula, and T (n) denotes the time needed to compute the transitive closure of a directed graph of n vertices. Furthermore, we investigate a tractable aspect of Horn and dual Horn relations.