Planar 3-SAT with a Clause/Variable Cycle
In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In...
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| Published in: | Discrete mathematics and theoretical computer science Vol. 21 no. 3; no. Discrete Algorithms |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
05.06.2019
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| Subjects: | |
| ISSN: | 1365-8050, 1365-8050 |
| Online Access: | Get full text |
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| Summary: | In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In the course of this research, different restrictions on the incidence graph of the formula have been devised, for which the problem also remains hard. In this paper, we investigate the restriction in which we require that the incidence graph can be augmented by the edges of a Hamiltonian cycle that first passes through all variables and then through all clauses, in a way that the resulting graph is still planar. We show that the problem of deciding satisfiability of a 3-SAT formula remains NP-complete even if the incidence graph is restricted in that way and the Hamiltonian cycle is given. This complements previous results demanding cycles only through either the variables or clauses. The problem remains hard for monotone formulas, as well as for instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of Planar 3-SAT with exactly three distinct variables per clause are always satisfiable, thus settling the question by Darmann, D\"ocker, and Dorn on the complexity of this problem variant in a surprising way.
Comment: Implementing style of DMTCS journal |
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| ISSN: | 1365-8050 1365-8050 |
| DOI: | 10.23638/DMTCS-21-3-18 |