Thresholds of Random k-Sat

Years and Authors of Summarized Original Work

• 2002; Kaporis, Kirousis, Lalas

Problem Definition

Consider n Boolean variables $V = \{x_{1},\ldots,x_{n}\}$ and the corresponding set of 2n literals $L =\{ x_{1}\overline{x}_{1}\ldots,x_{n},\overline{x}_{n}\}$. A k-clause is a disjunction of k literals of distinct underlying variables. A random formula ϕ_n, m in k conjunctive normal form (k-CNF) is the conjunction of m clauses, each selected in a uniformly random and independent way among the $2^{k}\left (\begin{array}{c}n \\ \\ k\end{array}\right )$ possible k-clauses on n variables in V. The density r _k of a k-CNF formula ϕ_n, m is the clauses-to-variables ratio m/n.

It was conjectured that for each k ≥ 2 there exists a critical density r _k ^∗ such that asymptotically almost all (a.a.a.) k-CNF formulas with density r < r _k ^∗ (r > r _k ^∗) are satisfiable (unsatisfiable, respectively). So far, the conjecture has been proved only for k = 2 [3, 11]. For k≥ 3, the conjecture still remains...