The Local Lemma is Tight for SAT
Heidi Gebauer, Tibor Szabó, Gábor Tardos · 2011
We construct unsatisfiable k-CNF formulas where every clause has k distinct literals and every variable appears in at most ( 2 e + o(1)) 2 k clauses. The lopsided Local Lemma k shows that our result is asymptotically best possible: every k-CNF formula where every variable appears in at most 2 k+1 − 1 clauses is satisfiable. The determination of this e(k+1) 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 YESinstance 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)) 2 k. 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. 1