A probabilistic study of generalized solution concepts in satisfiability testing and constraint programming

Zhang, Peng; Gao, Yong

doi:10.1016/j.tcs.2016.04.041

Cited by 8 publications

(5 citation statements)

References 16 publications

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“…In addition, the set of (1, 0)-super solutions is a subset of (1, 1)-super solutions, and the set of (1, 1)-super solutions is a subset of standard solutions. This explains why the upper bound on r of having (1, 0)-super solutions in Theorem 5 in [9] is smaller than that of having (1, 1)-super solutions in Theorem 1 and standard solutions obtained in [6].…”

Section: Introductionmentioning

confidence: 95%

“…Let σ be a fixed assignment, I be a random instance of model RB, and σ |= I be that σ satisfies I. We consider the following events which was similarly used in [9]:…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In [9], Zhang and Gao studied the (1, 0)-super solutions of model RB where they consider the special case k = 2 and d = √ n. By using the first moment method, they derived an upper bound of the threshold of having a (1, 0)-super solution asymptotically with probability 1, and established a condition for the expected number of super solutions to grow exponentially. (1, 0)-super solutions have also been studied in random (3+p)-SAT [12].…”

Section: Introductionmentioning

confidence: 99%

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Super Solutions of the Model RB

Zhou¹,

Wei²

2021

Preprint

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Section: Introductionmentioning

confidence: 95%

“…Let σ be a fixed assignment, I be a random instance of model RB, and σ |= I be that σ satisfies I. We consider the following events which was similarly used in [9]:…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Super Solutions of the Model RB

Zhou¹,

Wei²

2021

Preprint

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“…The problem to determine if a CNF formula is (1,0)-satisfiable is represented as (1,0)-SAT, and this corresponds to a k-CNF formula. It was shown in [22] that, for k ≤ 3, (1,0)-k-SAT is in P; otherwise it is in NP-complete. They also proved that, for Constrained Density (which is the clause-tovariable ratio) α < 1/3, a random 3-CNF formula is (1,0)-satisfiable with high probability, and not (1,0)-satisfiable with high probability for α > 1/3.…”

Section: Related Workmentioning

confidence: 99%

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

Wang

Liu

et al. 2022

Electronics

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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).

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“…(1,0)- k -SAT is the special version of (1,0)-SAT where each clause has exactly k distinct literals. Zhang P in [ 9 ] proved that (1,0)- k -SAT is in P for

and in NP-complete for

. Besides, it was proved that a random (1,0)-3-SAT are satisfiable with high probability for Constrained Density (the clause-to-variable ratio)

, and not satisfiable with high probability for

.…”

Section: Related Workmentioning

confidence: 99%

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

Wang

2020

Entropy

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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 ) .

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A probabilistic study of generalized solution concepts in satisfiability testing and constraint programming

Cited by 8 publications

References 16 publications

Super Solutions of the Model RB

Super Solutions of the Model RB

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

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

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