Bounds on Threshold of Regular Random $k$-SAT

We consider the regular model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad et. al. We derive an upper bound on the satisfiability threshold and NAE-satisfiability threshold for regular random $k$-SAT for any $k \geq 3$. We show that these bounds matches with the corresponding bound for the uniform model of formula generation. We derive lower bound on the threshold by applying the second moment method to the number of satisfying assignments. For large $k$, we note that the obtained lower bounds on the threshold of a regular random formula converges to the lower bound obtained for the uniform model. Thus, we answer the question posed in \cite{AcM06} regarding the performance of the second moment method for regular random formulas.

💡 Research Summary

The paper investigates the satisfiability and Not‑All‑Equal (NAE) satisfiability thresholds for the regular random k‑SAT model, a variant of random CNF generation introduced by Boufkhad et al. In this model each variable appears in exactly d literals and each clause contains exactly k literals, so the incidence matrix of variables versus clauses is regular. The authors first apply the first‑moment method to the number of satisfying assignments X. By computing E