Function problem

A function problem is defined by a relation over strings of an arbitrary alphabet :

Note that does not have to be a functional binary relation.

An algorithm solves if for every input such that there exists a satisfying , the algorithm produces one such , and if there are no such , it rejects.

A promise function problem permits the algorithm to do anything (thus may not terminate) if no such exists.

A well-known function problem is given by the functional Boolean satisfiability problem, FSAT for short. The problem, which is closely related to the SAT decision problem, can be formulated as follows:

Given a propositional formula with variables , find an assignment such that evaluates to or decide that no such assignment exists.

In this case the relation is given by pairs of suitably encoded propositional formulas and satisfying assignments. While a SAT algorithm, fed with a formula , only needs to return "unsatisfiable" or "satisfiable", an FSAT algorithm needs to return some satisfying assignment in the latter case.

Other notable examples include the travelling salesman problem, which asks for the route taken by the salesman, and the integer factorization problem, which asks for the list of factors.

Consider an arbitrary decision problem in the class NP. By the definition of NP, there is a system of certificates such that each problem instance that is answered 'yes' has a polynomial-size certificate that serves as a proof for the 'yes' answer (and problem instances answered 'no' have no such certificates). Thus, the set of these pairs forms a relation, representing the function problem "given in , find a certificate for ". This function problem is called a function variant of ; it belongs to the class FNP.

Conversely, every problem R in FNP induces a (unique) corresponding decision problem: given x, decide if there exists some y such that R(x,y) holds.

FNP can be thought of as the function class analogue of NP, in that solutions of FNP problems can be efficiently (i.e., in polynomial time in terms of the length of the input) verified, but not necessarily efficiently found. In contrast, the class FP, which can be thought of as the function class analogue of P, consists of function problems for which solutions can be found in polynomial time.

Observe that the problem FSAT introduced above can be solved using only polynomially many calls to a subroutine that decides the SAT problem: An algorithm can first ask whether the formula is satisfiable. After that the algorithm can fix variable to TRUE and ask again. If the resulting formula is still satisfiable the algorithm keeps fixed to TRUE and continues to fix , otherwise it decides that has to be FALSE and continues. Thus, FSAT is solvable in polynomial time using an oracle deciding SAT. In general, a problem in FNP is called self-reducible if it can be solved in polynomial time using an oracle for its induced decision problem. Every function variant of every NP-complete problem is self-reducible. There are several (slightly different) notions of self-reducibility.

Function problems can be reduced much like decision problems: Given function problems and we say that reduces to if there exist polynomially-time computable functions and such that for all instances of and possible solutions of , it holds that

• If has an -solution, then has an -solution.
• 

It is therefore possible to define FNP-hard problems analogous to NP-hard problems:

A problem is FNP-hard if every problem in FNP can be reduced to . A problem is FNP-complete if it is FNP-hard and in FNP. The problem FSAT is an FNP-complete problem, and hence by self-reducibility of FSAT it holds that if and only if .

The relation used to define function problems has the drawback of being possibly incomplete: Not every input necessarily has a counterpart such that . Therefore the question of computability of outputs is not separated from the question of their existence. To overcome this problem it is convenient to consider the restriction of function problems to total relations yielding the class TFNP as a subclass of FNP. This class contains problems such as the computation of pure Nash equilibria in certain strategic games where a solution is guaranteed to exist. In addition, if TFNP contains any FNP-complete problem it follows that .