Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
On subclasses of minimal unsatisfiable formulas
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF
Annals of Mathematics and Artificial Intelligence
An Application of Matroid Theory to the SAT Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Lean clause-sets: generalizations of minimally unsatisfiable clause-sets
Discrete Applied Mathematics - The renesse issue on satisfiability
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
Constraint Satisfaction Problems in Clausal Form I: Autarkies and Deficiency
Fundamenta Informaticae
On davis-putnam reductions for minimally unsatisfiable clause-sets
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
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We consider the question of the existence of variables with few occurrences in boolean conjunctive normal forms (clause-sets). Let µvd(F) for a clause-set F denote the minimal variable-degree, the minimum of the number of occurrences of variables. Our main result is an upper bound µvd(F) = nM(σ(F)) = σ(F) + 1 + log2(σ(F)) for lean clause-sets F in dependency on the surplus σ(F). Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets. For the surplus we have σ(F) = σ(F) = c(F)-n(F), using the deficiency σ(F) of clause-sets, the difference between the number of clauses and the number of variables. nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of natural numbers all numbers of the form 2n-1. As an application of the upper bound we obtain that clausesets F violating µvd(F) = nM(σ(F)) must have a non-trivial autarky (so clauses can be removed satisfiability-equivalently by an assignment satisfying some clauses and not touching the other clauses). It is open whether such an autarky can be found in polynomial time.