A simplified NP-complete satisfiability problem

Cook [l] has shown that 3-SAT, the Boolean satisfiability problem restricted to instances with exactly three variables per clause, is NP-complete. This is a tightest possible restriction on the number of variables in a clause because as Even et al. [2] demonstrate, 2-SAT is in P. Horowitz and Sahni [5] point up the importance of finding the strongest possible restrictions under which a problem remains NPcomplete. First, this can help clarify the interesting boundary between problems known to be in P and those that are not. Second, it can make it easier to establish the NP-completeness of new problems by allowing easier transformations. (For a comprehensive treatment of the subject, see [3].) To prove the Euclidean travelling salesman problem NP-hard, Papadimitriou [6] first reduces 3-SAT to 3-SAT where each variable appears in at most five clauses. The question arises, are any further reductions in this direction possible? In this note we show that 3-SAT remains NP-complete even when each variable appears at most four times. Let r,s-SAT denote the class of instances with exactly r variables per clause and at most s occurrences per variable. We prove the 3,4-SAT result to be the strongest possible and show that 3,3-SAT is in fact trivial. In addition we show that the Boolean satisfiability problem is solvable in linear time if no variable appears more than twice, regardless of the number of variables per clause. All Boolean expressions are taken to be in conjunctive normal form with no repeated variables in a clause.