Solving SAT for CNF Formulas with a One-Sided Restriction on Variable Occurrences

In this paper we consider the class of boolean formulas in Conjunctive Normal Form (CNF) where for each variable all but at most d occurrences are either positive or negative. This class is a generalization of the class of CNF formulas with at most d occurrences (positive and negative) of each variable which was studied in [Wahlström, 2005]. Applying complement search [Purdom, 1984], we show that for every d there exists a constant γd < 2 − 1 2d+1 such that satisfiability of a CNF formula on n variables can be checked in runtime O(γ d ) if all but at most d occurrences of each variable are either positive or negative. We thoroughly analyze the proposed branching strategy and determine the asymptotic growth constant γd more precisely. Finally, we show that the trivial O(2) barrier of satisfiability checking can be broken even for a more general class of formulas, namely formulas where the positive or negative literals of every variable have what we will call a d–covering. To the best of our knowledge, for the considered classes of formulas there are no previous non-trivial upper bounds on the complexity of satisfiability checking.