On the Boolean Connectivity Problem for Horn Relations

Gopalan et al. studied in ICALP06 [17] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on the connectivity problems in Schaefer’s framework. 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 there exists a set S of Horn relations such that the connectivity problem for S is coNP-complete. 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 with respect to characteristic sets.