Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
On the r,s satisfiability problem and a conjecture of Tovey
Discrete Applied Mathematics
Lopsided Lova´sz Local Lemma and Latin transversals
ARIDAM III Selected papers on Third advanced research institute of discrete applied mathematics
Coding and information theory
One more occurrence of variables makes satisfiability jump from trivial to NP-complete
SIAM Journal on Computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
DNF tautologies with a limited number of occurences of every variable
Theoretical Computer Science
Improved bounds and algorithms for hypergraph 2-coloring
Random Structures & Algorithms
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
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
On the structure of some classes of minimal unsatisfiable formulas
Discrete Applied Mathematics - The renesse issue on satisfiability
Homomorphisms of conjunctive normal forms
Discrete Applied Mathematics - The renesse issue on satisfiability
Computing unsatisfiable k-SAT instances with few occurrences per variable
Theoretical Computer Science
A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable
SIAM Journal on Discrete Mathematics
The Lovász Local Lemma and Satisfiability
Efficient Algorithms
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
A kolmogorov complexity proof of the lovász local lemma for satisfiability
COCOON'11 Proceedings of the 17th annual international conference on Computing and combinatorics
Journal of the ACM (JACM)
A Kolmogorov complexity proof of the Lovász Local Lemma for satisfiability
Theoretical Computer Science
Constraint satisfaction, packet routing, and the lovasz local lemma
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most (2/e +o(1)) 2k/k clauses. The lopsided Local Lemma shows that our result is asymptotically best possible: every k-CNF formula where every variable appears in at most 2k+1/e(k+1) − 1 clauses is satisfiable. The determination of this extremal function is particularly important as it represents the value where the k-SAT problem exhibits its complexity hardness jump: from having every instance being a YES-instance it becomes NP-hard just by allowing each variable to occur in one more clause. The asymptotics of other related extremal functions are also determined. Let l(k) denote the maximum number, such that every k-CNF formula with each clause containing k distinct literals and each clause having a common variable with at most l(k) other clauses, is satisfiable. We establish that the bound on l(k) obtained from the Local Lemma is asymptotically optimal, i.e., l(k) = (1/e + o(1)) 2k. The constructed formulas are all in the class MU(1) of minimal unsatisfiable formulas having one more clause than variables and thus they resolve these asymptotic questions within that class as well. The SAT-formulas are constructed via the binary trees of [10]. In order to construct the trees a continuous setting of the problem is defined, giving rise to a differential equation. The solution of the equation diverges at 0, which in turn implies that the binary tree obtained from the discretization of this solution has the required properties.