Revision #1 Authors: Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto

Accepted on: 23rd March 2007 00:00

Downloads: 3206

Keywords:

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.

TR07-029 Authors: Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto

Publication: 22nd March 2007 16:58

Downloads: 2408

Keywords:

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.