Computational complexity of some restricted instances of 3-SAT

Tovey [A simplified satisfiability problem, DiscreteAppl. Math. 8 (1984) 85-891 showed that it is NP-hard to decide the satisfiability of 3-SAT instances in which every variable occurs four times, while every instance of 3-SAT in which each variable occurs three times is satisfiable. We explore the border between these two problems. Answering a question of Iwarna and Takaki, we show that, for every fixed k >= 0, there is a polynomial-time algorithm to determine the satisfiability of 3-SAT instances in which k variables occur four times and the remaining variables occur three times. On the other hand, it is NP-hard to decide the satisfiability of 3-SAT instances in which all but one variable occurs three times, and the remaining variable is allowed to occur an arbitrary number of times. (c) 2006 Elsevier B.V. All rights reserved.