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| In [[computational complexity theory]], the [[complexity class]] '''TFNP''' is a subclass of [[FNP (complexity)|FNP]] where a solution is guaranteed to exist. It stands for "Total Function Nondeterministic Polynomial."
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| :A binary relation P(''x'',''y'') is in '''TFNP''' if and only if there is a deterministic polynomial time algorithm that can determine whether P(''x'',''y'') holds given both ''x'' and ''y'', and for every ''x'', there exists a ''y'' such that P(''x'',''y'') holds.
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| The job of an ''TFNP'' algorithm is to establish, given an ''x'' give one possible value for a ''y'' such that P(''x'',''y'') holds.
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| [[FP (complexity)|FP]] is a subclass of TFNP. TFNP also contains subclasses [[PLS (complexity)|PLS]], [[PPA (complexity)|PPA]], [[PPAD (complexity)|PPAD]], and [[PPP (complexity)|PPP]].
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| TFNP = FP would imply [[P (complexity)|P]] = [[NP (complexity)|NP]] <math>\cap</math> [[Co-NP|coNP]], and therefore that factoring and simplex pivoting are in P.
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| In contrast to FNP, there is no known recursive enumeration of machines for problems in TFNP. It seems that such classes, and therefore TFNP, do not have complete problems.
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| == References ==
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| * Megiddo and Papadimitriou. [http://citeseer.ist.psu.edu/megiddo89note.html A Note on Total Functions, Existence Theorems and Computational Complexity (1989)].
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| [[Category:Complexity classes]]
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Latest revision as of 02:40, 12 September 2014
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