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A variant of the 3-satisfiability problem is the one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT). Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has exactly one TRUE literal (and thus exactly two ...
The NP-completeness of NAE3SAT can be proven by a reduction from 3-satisfiability (3SAT). [2] First the nonsymmetric 3SAT is reduced to the symmetric NAE4SAT by adding a common dummy literal to every clause, then NAE4SAT is reduced to NAE3SAT by splitting clauses as in the reduction of general -satisfiability to 3SAT.
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-completeness. [1] [3] The satisfiability of a circuit containing arbitrary binary gates can be decided in time ().
For example, the 3-satisfiability problem, a restriction of the Boolean satisfiability problem, remains NP-complete, whereas the slightly more restricted 2-satisfiability problem is in P (specifically, it is NL-complete), but the slightly more general max. 2-sat. problem is again NP-complete.
For every ε > 0, there is a PCP-verifier M for 3-SAT that reads a random string r of length ( ()) and computes query positions i r, j r, k r in the proof π and a bit b r. It accepts if and only if 'π(i r) ⊕ π(j r) ⊕ π(k r) = b r. The Verifier has completeness (1−ε) and soundness 1/2 + ε (refer to PCP (complexity)). The ...
This proof is based on the one given by Garey & Johnson 1979, pp. 38–44, Section 2.6. There are two parts to proving that the Boolean satisfiability problem (SAT) is NP-complete. One is to show that SAT is an NP problem.
Special cases of Schaefer's dichotomy theorem include the NP-completeness of SAT (the Boolean satisfiability problem) and its two popular variants 1-in-3 SAT and not-all-equal 3SAT (often denoted by NAE-3SAT). In fact, for these two variants of SAT, Schaefer's dichotomy theorem shows that their monotone versions (where negations of variables ...
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