On simplified NP-complete variants of Monotone[formula omitted]-Sat

Abstract

We consider simplified versions of 3-Sat, the variant of the famous Satisfiability Problem where each clause is made up of exactly three distinct literals formed over pairwise distinct variables. More precisely, the focus of this work is laid on Monotone3-Sat, the restriction of 3-Sat to formulas with monotone clauses, where a clause is monotone if it contains only unnegated variables or only negated variables. We prove several hardness results for Monotone3-Sat with respect to a variety of restrictions imposed on the variable appearances.In particular, we show that for any k≥2, Monotone3-Sat turns out to be NP-complete even if each variable appears exactly k times unnegated and exactly k times negated. Therewith, for Monotone3-Sat with balanced variable appearances we establish a sharp boundary between NP-complete and polynomial time solvable cases.In addition, we prove that for any k≥5, Monotone3-Sat is NP-complete even if each variable appears exactly k times unnegated and exactly once negated. Further, we prove that the problem remains NP-complete when restricted to instances in which each variable appears either exactly once unnegated and three times negated or the other way around. Thereby, we improve on a result by Darmann et al. (2018) showing NP-completeness for four appearances per variable. Our stronger result also implies that 3-Sat remains NP-complete even if each variable appears exactly three times unnegated and once negated, therewith complementing a result by Berman et al. (2003).