Linear CNF formulas and satisfiability

Abstract

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 of linear CNF classes

“…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.…”

“…SAT can be decided in polynomial time for exact linear formulas [1]. We show that NAE-SAT and even its counting version are also polynomial-time decidable restricted to monotone exact linear formulas relying on a result in [2].…”

“…Clauses of linear formulas overlap only sparsely in the sense that they can have at most one variable in common. Various aspects of SAT concerning linear formulas have been investigated recently in [1,7]. Linear formulas deserve interest from several perspectives.…”

The Lovász Local Lemma and Satisfiability

“…In the case of linear formulas, explicit constructions have been given in [34,35]. However, the size of the constructed formulas is terrifying: For an unsatisfiable linear k-CNF formula, it takes 2 2 .…”

“…The extremal examples for most parameters considered so far are in MU(1): for f , l and lc via the tree derived formulas from Section 3; also an unsatisfiable k-CNF formula with 2 k clauses can be found in MU(1) (see argument for Proposition 1 below). For linear CNF formulas, the explicit constructions in [34,35] can be adapted to result in formulas in MU(1), but they exhibit the tremendous size as mentioned. This is inherently so for linear CNF formulas in MU(1).…”