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CircuitSAT is closely related to Boolean satisfiability problem (SAT), and likewise, has been proven to be NP-complete. [2] It is a prototypical NP-complete problem; the Cook–Levin theorem is sometimes proved on CircuitSAT instead of on the SAT, and then CircuitSAT can be reduced to the other satisfiability problems to prove their NP ...
3-satisfiability can be generalized to k-satisfiability (k-SAT, also k-CNF-SAT), when formulas in CNF are considered with each clause containing up to k literals. [citation needed] However, since for any k ≥ 3, this problem can neither be easier than 3-SAT nor harder than SAT, and the latter two are NP-complete, so must be k-SAT.
NAE3SAT remains NP-complete when all clauses are monotone (meaning that variables are never negated), by Schaefer's dichotomy theorem. [3] Monotone NAE3SAT can also be interpreted as an instance of the set splitting problem , or as a generalization of graph bipartiteness testing to 3-uniform hypergraphs : it asks whether the vertices of a ...
The use of SAT to prove the existence of an NP-complete problem can be extended to other computational problems in logic, and to completeness for other complexity classes. The quantified Boolean formula problem (QBF) involves Boolean formulas extended to include nested universal quantifiers and existential quantifiers for its variables.
The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it. Therefore, it is useful to know a variety of NP-complete problems. The list below contains some well-known problems that are NP-complete when expressed as decision problems.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
The decision version of MAX-3SAT is NP-complete. Therefore, a polynomial-time solution can only be achieved if P = NP. An approximation within a factor of 2 can be achieved with this simple algorithm, however: Output the solution in which most clauses are satisfied, when either all variables = TRUE or all variables = FALSE.
That is, efficiently constructible circuits for SAT would lead to a stronger collapse, P = NP. The assumption of the Karp–Lipton theorem, that these circuits exist, is weaker. But it is still possible for an algorithm in the complexity class Σ 2 {\displaystyle \Sigma _{2}} to guess a correct circuit for SAT.