(1,0)-Super Solutions of (k,s)-CNF Formula

Abstract

A (1,0)-super solution is a satisfying assignment such that if the value of any one variable is flipped to the opposite value, the new assignment is still a satisfying assignment. Namely, every clause must contain at least two satisfied literals. Because of its robustness, super solutions are concerned in combinatorial optimization problems and decision problems. In this paper, we investigate the existence conditions of the (1,0)-super solution of ( k , s ) -CNF formula, and give a reduction method that transform from k-SAT to (1,0)- ( k + 1 , s ) -SAT if there is a ( k + 1 , s )-CNF formula without a (1,0)-super solution. Here, ( k , s ) -CNF is a subclass of CNF in which each clause has exactly k distinct literals, and each variable occurs at most s times. (1,0)- ( k , s ) -SAT is a problem to decide whether a ( k , s ) -CNF formula has a (1,0)-super solution. We prove that for k > 3 , if there exists a ( k , s ) -CNF formula without a (1,0)-super solution, (1,0)- ( k , s ) -SAT is NP-complete. We show that for k > 3 , there is a critical function φ ( k ) such that every ( k , s ) -CNF formula has a (1,0)-super solution for s ≤ φ ( k ) and (1,0)- ( k , s ) -SAT is NP-complete for s > φ ( k ) . We further show some properties of the critical function φ ( k ) .

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

“…Therefore, we get ϕ(k, d) ≥ ϕ(k). Because the critical function ϕ(k) is an increasing function, proved in [27], we obtain ϕ(k, d) ≥ ϕ(k − 1). Any d-regular (k, s)-CNF formula obviously should be a (d + 1)-regular (k, s)-CNF formula, so we get ϕ(k, d) ≥ ϕ(k, d + 1).…”

“…In [12], we proved that d-regular (k, s)-SAT also has the Transition Phenomenon from triviality (output the affirmative answer without computation) to NPcompleteness, and gave some favorable properties of the critical function f (k, d). In [27], we gave some existence conditions of a (1,0)-super solution, and pointed out that if there is an unsatisfiable (1,0)-(k, s)-SAT instance, then (1,0)-(k, s)-SAT problem is NP-complete for k > 3. That paper shows that the (1,0)-(k, s)-SAT problem also exhibits the Transition Phenomenon.…”

“…Therefore, if ϕ(k) < s, then ϕ(k, s − 2) < s. For d = ϕ(k) − 1, we get ϕ(k, d) = ϕ(k). According to some properties of the critical function f (k, d) and ϕ(k) presented in [12,27], we can obtain the following corollaries. Proof.…”

Uniquely Satisfiable d-Regular (k,s)-SAT Instances

“…([ 17 ]) . If the representation matrix of a formula F is then the formula is satisfiable and every satisfying assignment forces all variables to a same value.…”

“…In order to further study SAT problems with regular structures, we introduced d -regular ( )-CNF formula in [ 16 , 17 ]. The regular ( )-CNF formula requires that each clause contains exactly k variables and each variable occurs in exactly s clauses.…”