Bounds on Threshold of Regular Random k-SAT

Abstract

We consider the regular model of formula generation in conjunctive normal form (CNF) introduced by Boufkhad et. al. in [6]. In [6], it was shown that the threshold for regular random 2-SAT is equal to unity. Also, upper and lower bound on the threshold for regular random 3-SAT were derived. Using the first moment method, we derive an upper bound on the threshold for regular random k-SAT for any k ≥ 3 and show that for large k the threshold is upper bounded by 2^k ln (2). We also derive upper bounds on the threshold for Not-All-Equal (NAE) satisfiability for k ≥ 3 and show that for large k, the NAE-satisfiability threshold is upper bounded by 2^k − 1 ln (2). For both satisfiability and NAE-satisfiability, the obtained upper bound matches with the corresponding bound for the uniform model of formula generation [9,1].

For the uniform model, in a series of break through papers Achlioptas, Moore, and Peres showed that a careful application of the second moment method yields a significantly better lower bound on threshold as compared to any rigorously proven algorithmic bound [3,1]. The second moment method shows the existence of a satisfying assignment with uniform positive probability (w.u.p.p.). Thanks to the result of Friedgut for uniform model [10], existence of a satisfying assignment w.u.p.p. translates to existence of a satisfying assignment with high probability (w.h.p.). Thus, the second moment method gives a lower bound on the threshold. As there is no known Friedgut type result for regular random model, we assume that for regular random model existence of a satisfying assignments w.u.p.p. translates to existence of a satisfying assignments w.h.p. We derive the second moment of the number of satisfying assignments for regular random k-SAT for k ≥ 3. There are two aspects in deriving the lower bound using the second moment method. The first aspect is given any k, numerically evaluate the lower bound on the threshold. The second aspect is to derive the lower bound as a function of k for large enough k. We address the first aspect and evaluate the lower bound on threshold. The numerical evaluation suggests that as k increases the obtained lower bound on the satisfiability threshold of a regular random formula converges to the lower bound obtained for the uniform model. Similarly, we obtain lower bounds on the NAE-satisfiability threshold of the regular random formulas and observe that the obtained lower bound seems to converge to the corresponding lower bound for the uniform model as k increases.