FNP (complexity)

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In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is a bit of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains:

A binary relation P(x,y) is in FNP if and only if there is a deterministic polynomial time algorithm that can determine whether P(x,y) holds given both x and y.

The job of an FNP algorithm is to establish, given an x, whether there exists a y such that P(x,y) holds, and if so, give one possible value for that y. See FP for an explanation of the distinction between FP and FNP. There is an NP language directly corresponding to every FNP relation, sometimes called the decision problem induced by or corresponding to said FNP relation. It is the language formed by taking all the x for which P(x,y) holds given some y; however, there may be more than one FNP relation for a particular decision problem.

Many problems in NP, including many NP-complete problems, ask whether a particular object exists, such as a satisfying assignment, a graph coloring, or a clique of a certain size. The FNP versions of these problems ask not only if it exists but what its value is if it does. This means that the FNP version of every NP-complete problem is NP-hard. Bellare and Goldwasser showed in 1991 using some standard assumptions that there exist NP-complete problems such that their FNP versions are not self-reducible, implying that they are harder than their corresponding decision problem

FP = FNP if and only if P = NP.

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