The Transition Phenomenon of (1,0)-d-Regular (k, s)-SAT

Abstract

For a d-regular (k,s)-CNF formula, a problem is to determine whether it has a (1,0)-super solution. If so, it is called (1,0)-d-regular (k,s)-SAT. A (1,0)-super solution is an assignment that satisfies at least two literals of each clause. When the value of any one of the variables is flipped, the (1,0)-super solution is still a solution. Super solutions have gained significant attention for their robustness. Here, a d-regular (k,s)-CNF formula is a special CNF formula with clauses of size exactly k, in which each variable appears exactly s-times, and the absolute frequency difference between positive and negative occurrences of each variable is at most a nonnegative integer d. Obviously, the structure of a d-regular (k,s)-CNF formula is much more regular than other formulas. In this paper, we certify that, for k≥5, there is a critical function φ(k,d) such that, if s≤φ(k,d), all d-regular (k,s)-CNF formulas have a (1,0)-super solution; otherwise (1,0)-d-regular (k,s)-SAT is NP-complete. By the Lopsided Local Lemma, we get an existence condition of (1,0)-super solutions and propose an algorithm to find the lower bound of φ(k,d).