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SAT was the first problem known to be NP-complete, as proved by Stephen Cook at the University of Toronto in 1971 [8] and independently by Leonid Levin at the Russian Academy of Sciences in 1973. [9] Until that time, the concept of an NP-complete problem did not even exist.
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.
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 ...
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.
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 ...
Boolean satisfiability problem (SAT). [2] [3]: LO1 There are many variations that are also NP-complete. An important variant is where each clause has exactly three literals (3SAT), since it is used in the proof of many other NP-completeness results. [3]: p. 48 Circuit satisfiability problem; Conjunctive Boolean query [3]: SR31
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 ...
This counting problem is #P-complete, [35] which implies that it is not solvable in polynomial time unless P = NP. Moreover, there is no fully polynomial randomized approximation scheme for #2SAT unless NP = RP and this even holds when the input is restricted to monotone 2-CNF formulas, i.e., 2-CNF formulas in which each literal is a positive ...