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We investigate the computational complexity of the exact satisfiability problem (XSAT) restricted to certain subclasses of linear CNF formulas. These classes are defined through restricting the number of occurrences of variables and are therefore interesting because the complexity status does not follow from Schaefer΄s theorem [14,7]. Specifically we prove that XSAT remains NP-complete for linear formulas which are monotone and all variables occur exactly l times. We also present some complexity results for exact linear formulas left open in [9]. Concretely, we show that XSAT for this class is NP-complete, in contrast to SAT or NAE-SAT. This can also be established when clauses have length at least k, for fixed integer k ≥ 3. However, the XSAT-complexity for exact linear formulas with clause length exactly k remains open, but we provide its polynomial-time behaviour at least for every positive integer k ≤ 6. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]