Note: Computational complexity of some restricted instances of 3-SAT

Authors:
Piotr Berman;Marek Karpinski;Alexander D. Scott
Affiliations:
Department of Computer Science and Engineering, Pennsylvania State University, USA;Department of Computer Science, University of Bonn, Germany;Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
Venue:
Discrete Applied Mathematics
Year:
2007

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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, 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.