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 γ d <2 ? 1/2d+1 such that satisfiability of a CNF formula on n variables can be checked in runtime O(γ_dⁿ)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 γ _d more precisely. Finally, we show that the trivial O(2 ⁿ)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.