In computational complexity theory, a function problem is a computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem. For function problems, the output is not simply 'yes' or 'no'.

A function problem ${\displaystyle P}$ is defined by a relation ${\displaystyle R}$ over strings of an arbitrary alphabet ${\displaystyle \Sigma }$:

An algorithm solves ${\displaystyle P}$ if for every input ${\displaystyle x}$ such that there exists a ${\displaystyle y}$ satisfying ${\displaystyle (x,y)\in R}$, the algorithm produces one such ${\displaystyle y}$, and if there are no such ${\displaystyle y}$, it rejects.

A promise function problem permits the algorithm to do anything (thus may not terminate) if no such ${\displaystyle y}$ 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:

In this case the relation ${\displaystyle R}$ is given by pairs of suitably encoded boolean formulas and satisfying assignments. While a SAT algorithm, fed with a formula ${\displaystyle \varphi }$, 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 ${\displaystyle L}$ in the class NP. By the definition of NP, each problem instance ${\displaystyle x}$ that is answered 'yes' has a polynomial-size certificate ${\displaystyle y}$ which serves as a proof for the 'yes' answer. Thus, the set of these pairs ${\displaystyle (x,y)}$ forms a relation, representing the function problem "given ${\displaystyle x}$ in ${\displaystyle L}$, find a certificate ${\displaystyle y}$ for ${\displaystyle x}$". This function problem is called the function variant of ${\displaystyle L}$; it belongs to the class FNP.