Computational complexity of some restricted instances of 3-SAT

Piotr Berman, Marek Karpinski, Alexander D. Scott

Abstract: Tovey [A simplified satisfiability problem, Discrete Appl. Math. 8 (1984) 85–89] 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 Iwama and Takaki, we show that, for every fixed k⩾0<math><mi is="true">k</mi><mo is="true">⩾</mo><mn is="true">0</mn></math>, 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.