A short proof of Fisher's inequality
Discrete Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
New algorithms for exact satisfiability
Theoretical Computer Science
Linear CNF formulas and satisfiability
Discrete Applied Mathematics
A CNF class generalizing exact linear formulas
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
Complexity results for linear XSAT-Problems
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
Zero-One designs produce small hard SAT instances
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
XSAT and NAE-SAT of linear CNF classes
Discrete Applied Mathematics
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We investigate the computational complexity of some prominent variants of the propositional satisfiability problem (SAT), namely not-all-equal SAT (NAE-SAT) and exact SAT (XSAT) restricted to the class of linear conjunctive normal form (CNF) formulas. Clauses of a linear formula pairwise have at most one variable in common. We prove that NAE-SAT and XSAT both remain NP-complete when restricted to linear formulas. Since the corresponding reduction is not valid when input formulas are not allowed to have 2-clauses, we also prove that NAE-SAT and XSAT still behave NP-complete on formulas only containing clauses of length at least k, for each fixed integer k ≥ 3. Moreover, NP-completeness proofs for NAE-SAT and XSAT restricted to monotone linear formulas are presented. We also discuss the length restricted monotone linear formula classes regarding NP-completeness where a difficulty arises for NAE-SAT, when all clauses are k-uniform, for k ≥ 4. Finally, we show that NAE-SAT is polynomial-time decidable on exact linear formulas, where each pair of distinct clauses has exactly one variable in common. And, we give some hints regarding the complexity of XSAT on the exact linear class.