Abstract:
P. Gopalan, P. G. Kolaitis, E. N. Maneva and C. H. Papadimitriou studied in [Gopalan et al., ICALP2006] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on the connectivity problems in Schaefer's framework [Schaefer, STOC1978]. 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 $O(\min \{n|\gvp|, T(n)\})$ time, where $n$ denotes the number of variables, $\gvp$ 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.

Abstract:
P. Gopalan, P. G. Kolaitis, E. N. Maneva and C. H. Papadimitriou studied in [Gopalan et al., ICALP2006] connectivity properties of the solution-space of Boolean formulas, and investigated complexity issues on connectivity problems in Schaefer's framework [Schaefer, STOC1978]. 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 [Gopalan et al., ICALP 2006],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 |phi|, T(n)}) time, where n denotes the number of variables, phi 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.