One more occurrence of variables makes satisfiability jump from trivial to NP-complete
SIAM Journal on Computing
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
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable
SIAM Journal on Discrete Mathematics
New algorithms for exact satisfiability
Theoretical Computer Science
Color critical hypergraphs with many edges
Journal of Graph Theory
On Some SAT-Variants over Linear Formulas
SOFSEM '09 Proceedings of the 35th Conference on Current Trends in Theory and Practice of Computer Science
Linear CNF formulas and satisfiability
Discrete Applied Mathematics
Approximation algorithms and hardness results for the clique packing problem
Discrete Applied Mathematics
Algorithms for variable-weighted 2-SAT and dual problems
SAT'07 Proceedings of the 10th 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
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XSAT and NAE-SAT are important variants of the propositional satisfiability problem (SAT). XSAT contains all CNF formulas that can be satisfied by setting exactly one literal in each clause to 1, whereas NAE-SAT requires satisfying truth assignments not setting all literals equally in any clause. Both variants are studied here regarding their computational complexity of linear CNF formulas, whose clauses are allowed to have at most one variable in common. We prove that both variants remain NP-complete for (monotone) linear formulas, yielding the conclusion that also bicolorability of linear hypergraphs is NP-complete. The reduction used gives rise to the complexity investigations of several monotone linear subclasses of both variants that are parameterized by the size of clauses, or by the number of occurrences of variables. For particular values of those parameters, we are able to show the NP-completeness of XSAT and NAE-SAT, though we cannot provide a complete treatment. Finally, we focus on exact linear formulas, where clauses intersect pairwise, and for which SAT is known to be polynomial-time solvable. Here, we prove NP-completeness of XSAT for exact linear formulas. If in addition the clauses are required to be of uniform length, we show that both XSAT and NAE-SAT are decidable in polynomial time, what even holds for their counting versions.