Satisfiability Thresholds for k-CNF Formula with Bounded Variable Intersections

Abstract

We determine the thresholds for the number of variables, number of clauses, number of clause intersection pairs and the maximum clause degree of a k-CNF formula that guarantees satisfiability under the assumption that every two clauses share at most $α$ variables. More formally, we call these formulas $α$-intersecting and define, for example, a threshold $μ_i(k,α)$ for the number of clause intersection pairs $i$, such that every $α$-intersecting k-CNF formula in which at most $μ_i(k,α)$ pairs of clauses share a variable is satisfiable and there exists an unsatisfiable $α$-intersecting k-CNF formula with $μ_m(k,α)$ such intersections. We provide a lower bound for these thresholds based on the Lovasz Local Lemma and a nearly matching upper bound by constructing an unsatisfiable k-CNF to show that $μ_i(k,α) = \tildeΘ(2^{k(2+1/α)})$. Similar thresholds are determined for the number of variables ($μ_n = \tildeΘ(2^{k/α})$) and the number of clauses ($μ_m = \tildeΘ(2^{k(1+\frac{1}α)})$) (see [Scheder08] for an earlier but independent report on this threshold). Our upper bound construction gives a family of unsatisfiable formula that achieve all four thresholds simultaneously.