Abstract
In a regular (d,k)-CNF formula, each clause has length k and each variable appears d times. A regular structure such as this is symmetric, and the satisfiability problem of this symmetric structure is called the (d,k)-SAT problem for short. The regular exact 2-(d,k)-SAT problem is that for a (d,k)-CNF formula F, if there is a truth assignment T, then exactly two literals of each clause in F are true. If the formula F contains only positive or negative literals, then there is a satisfiable assignment T with a size of 2n/k such that F is 2-exactly satisfiable. This paper introduces the (d,k)-SAT instance generation model, constructs the solution space, and employs the method of the first and second moments to present the phase transition point d* of the 2-(d,k)-SAT instance with only positive literals. When d<d*, the 2-(d,k)-SAT instance can be satisfied with high probability. When d>d*, the 2-(d,k)-SAT instance can not be satisfied with high probability. Finally, the verification results demonstrate that the theoretical results are consistent with the experimental results.
Highlights
The phase transition points of the 2-(d, k)-SAT problem with symmetric structure are obtained by using the method of first and second moments
By constructing the solution space and using the methods of first and second moments, the satisfiable phase transition point d∗ for the 2-(d, k)-SAT problem with only positive literals was provided. d∗ is a function of k (k ≥ 3)
For a positive integer d, if d < d∗, the problem can be satisfied with high probability
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. For the (d, k)-SAT problem, if there is a set of variable assignments such that exactly two literals in each clause of the (d, k)-SAT instance are true, such problem is called the regular exact 2-(d, k)-SAT problem This problem is still hard to solve in the “worst” case, and the study of it is helpful to further analyze other SAT problems with symmetric structure. Boufkhad et al firstly proposed the regular (d, k)-SAT problem with this symmetric structure and pointed out that the satisfiable phase transition point of the random regular (3, k)-SAT problem was 2.46 ≤ αk,3 ≤ 3.78 in [11]. The author proposed the random regular exact cover problem (exactly one in every clause is true), studied its satisfiable phase transition, and proved it by using the subgraph method [16]. The verification results demonstrate that the theoretical results are in agreement with the experimental results
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