View source for Not-all-equal 3-satisfiability

In [[computational complexity]], '''not-all-equal 3-satisfiability''' ('''NAE3SAT''') is an [[NP-complete]] variant of the [[Boolean satisfiability problem]], often used in proofs of NP-completeness.{{r|moqo}}

==Definition==
Like [[3-satisfiability]], an instance of the problem consists of a collection of [[Boolean variable]]s and a collection of clauses, each of which combines three variables or negations of variables. However, unlike 3-satisfiability, which requires each clause to have at least one true Boolean value, NAE3SAT requires that the three values in each clause are not all equal to each other (in other words, at least one is true, and at least one is false).{{r|noc}}

==Hardness==
The NP-completeness of NAE3SAT can be proven by a [[reduction (complexity)|reduction]] from 3-satisfiability (3SAT).{{r|noc}} First the nonsymmetric 3SAT is reduced to the symmetric NAE4SAT by adding a common dummy literal <math>s</math> to every clause, then NAE4SAT is reduced to NAE3SAT by splitting clauses as in the reduction of general <math>k</math>-satisfiability to 3SAT.

In more detail, a 3SAT instance <math> \Phi = \bigwedge_{i=1}^m (l_{i,1} \vee l_{i,2} \vee l_{i,3}) </math> (where the <math> l_{i,j} </math> are arbitrary literals) is reduced to the NAE4SAT instance <math> \Psi = \bigwedge_{i=1}^m (l_{i,1}, l_{i,2}, l_{i,3}, s) </math> where <math> s </math> is a new variable. A satisfying assignment for <math>\Phi</math> becomes a satisfying assignment for <math>\Psi</math> by setting <math> s = 0 </math>. Conversely a satisfying assignment with <math> s = 0 </math> for <math>\Psi</math> must have at least one other literal true in each clause and thus be a satisfying assignment for <math>\Phi</math>. Finally a satisfying assignment with <math> s = 1 </math> for <math>\Psi</math> can because of symmetry of <math>0</math> and <math>1</math> be flipped to produce a satisfying assignment with <math> s = 0 </math>. 

NAE3SAT remains NP-complete when all clauses are monotone (meaning that variables are never negated), by [[Schaefer's dichotomy theorem]].{{r|schaefer}}
Monotone NAE3SAT can also be interpreted as an instance of the [[set splitting problem]], or as a generalization
of [[bipartite graph|graph bipartiteness]] testing to 3-uniform [[hypergraph]]s: it asks whether the vertices of a hypergraph can be [[graph coloring|colored]] with two colors so that no hyperedge is monochromatic. More strongly, it is NP-hard to find colorings of 3-uniform hypergraphs with any constant number of colors, even when a 2-coloring exists.{{r|drs}}

==Easy cases==
Unlike 3SAT, some variants of NAE3SAT in which graphs representing the structure of variables and clauses are [[planar graphs]] can be solved in [[polynomial time]]. In particular this is true when there exists a planar graph with one vertex per variable, one vertex per clause, an edge for each variable–clause incidence, and a [[cycle (graph theory)|cycle]] of edges connecting all the variable vertices.{{r|moret}}

==References==
{{reflist|refs=

<ref name=drs>{{citation
 | last1 = Dinur | first1 = Irit | author1-link = Irit Dinur
 | last2 = Regev | first2 = Oded | author2-link = Oded Regev (computer scientist)
 | last3 = Smyth | first3 = Clifford
 | issue = 5
 | journal = Combinatorica
 | mr = 2176423
 | pages = 519–535
 | title = The hardness of 3-uniform hypergraph coloring
 | volume = 25
 | year = 2005 | doi=10.1007/s00493-005-0032-4}}</ref>

<ref name=moret>{{citation
 | last = Moret | first = B. M. E. | authorlink = Bernard Moret
 | date = June 1988
 | doi = 10.1145/49097.49099
 | issue = 2
 | journal = [[ACM SIGACT News]]
 | pages = 51–54
 | title = Planar NAE3SAT is in P
 | volume = 19| s2cid = 17219595 }}</ref>

<ref name=moqo>{{harvtxt|Moret|1988}}: "Among published proofs of NP-completeness, one finds more reductions from 3-Satisfiability (3SAT for short) and its main variants, One- in-three-3SAT (1in3SAT) and Not-all-equal 3SAT (NAE3SAT), than from any other NP-complete problem."</ref>

<ref name=noc>{{citation
 | last1 = Moore | first1 = Cristopher | author1-link = Cristopher Moore
 | last2 = Mertens | first2 = Stephan
 | contribution = Symmetry-breaking and NAESAT
 | contribution-url = https://books.google.com/books?id=z4zMiZyAE1kC&pg=PA133
 | isbn = 9780199233212
 | pages = 133–138
 | publisher = Oxford University Press
 | title = The Nature of Computation
 | year = 2011}}</ref>

<ref name=schaefer>{{citation
 | last = Schaefer | first = Thomas J.
 | authorlink=Thomas Jerome Schaefer
 | contribution = The complexity of satisfiability problems
 | location = New York
 | mr = 521057
 | pages = 216–226
 | publisher = ACM
 | title = Proc. Tenth [[ACM Symposium on Theory of Computing]] (STOC '78)
 | year = 1978}}</ref>

}}

[[Category:NP-complete problems]]
[[Category:Satisfiability problems]]