Satisfiability Thresholds for k-CNF Formula with Bounded Variable Intersections

TL;DR

This paper establishes precise thresholds for satisfiability in bounded intersection k-CNF formulas, revealing how clause intersections and variable sharing influence satisfiability limits.

Contribution

It provides tight bounds for thresholds related to clause intersections, variables, and clauses in α-intersecting k-CNF formulas, combining probabilistic and constructive methods.

Findings

Threshold for clause intersection pairs: (2^{k(2+1/)}).

Threshold for variables: (2^{k/}).

Threshold for clauses: (2^{k(1+1/)}).

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 $\alpha$ variables. More formally, we call these formulas $\alpha$-intersecting and define, for example, a threshold $\mu_i(k,\alpha)$ for the number of clause intersection pairs $i$, such that every $\alpha$-intersecting k-CNF formula in which at most $\mu_i(k,\alpha)$ pairs of clauses share a variable is satisfiable and there exists an unsatisfiable $\alpha$-intersecting k-CNF formula with $\mu_m(k,\alpha)$ 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 $\mu_i(k,\alpha) = \tilde{\Theta}(2^{k(2+1/\alpha)})$. Similar thresholds are determined for the number of variables ($\mu_n = \tilde{\Theta}(2^{k/\alpha})$) and the number of clauses ($\mu_m = \tilde{\Theta}(2^{k(1+\frac{1}{\alpha})})$) (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.