a b s t r a c tIn this paper, we study linear CNF formulas generalizing linear hypergraphs under combinatorial and complexity theoretical aspects w.r.t. SAT. We establish NP-completeness of SAT for the unrestricted linear formula class, and we show the equivalence of NPcompleteness of restricted uniform linear formula classes w.r.t. SAT and the existence of unsatisfiable uniform linear witness formulas. On that basis we prove NP-completeness of SAT for uniform linear classes in a resolution-based manner by constructing large-sized formulas. Interested in small witness formulas, we exhibit some combinatorial features of linear hypergraphs closely related to latin squares and finite projective planes helping to construct rather dense, and significantly smaller unsatisfiable k-uniform linear formulas, at least for the cases k = 3, 4.

XSAT and NAE-SAT are important variants of the propositional satisfiability problem (SAT). Both are studied here regarding their computational complexity of linear CNF formulas. 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 investigation of both variants for several monotone linear subclasses that are parameterized by the size of clauses or by the number of occurrences of variables. In particular cases of these parameter values we are able to verify the NP-completeness of XSAT respectively 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 [1]. We verify the same assertion for NAE-SAT relying on a result in [2]; whereas we obtain NP-completeness for XSAT of exact linear formulas. The case of uniform clause size k remains open for the latter problem. However, we can provide its polynomial-time behavior for k at most 6.

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Let F = C 1 ∧ · · · ∧ C m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C i contains at most three literals l over V . Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2 0.18674n ). Thereby we improve a result in [2,3] stating X3SAT ∈ O(2 0.2072n ) and a bound of O(2 0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT ∈ O(2 0.16254n ).

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