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, that is, it isn't just YES or NO.
An algorithm solves if for every input the algorithm either produces an output such that or decides that no such exists.
A well-known function problem is given by the Functional Boolean Satisfyability Problem, FSAT for short. The problem, which is closely related to the SAT decision problem can be formulated as follows:
Given a boolean 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 tuples of suitably encoded boolean formulas and satisfying assignments.
Relationship to other complexity classes
Consider an arbitrary decision problem in the class NP. By definition each problem instance which are answered 'yes' have a certificate which servers as a proof for the 'yes' answer. Thus, the set of these tuples forms a relation. The complexity class derived from this transformation is denoted by or FNP for short. The mapping of the complexity class P is denoted by FP. The class FP is the set of function problems which can be solved by a deterministic Turing machine in polynomial time, whereas FNP is the set of function problems which can be solved by a non-deterministic Turing machine in polynomial time.
Observe that the problem FSAT introduced above can be solved using only polynomially many calls to a subroutine which 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 NP is called self-reducible if its function variant can be solved in polynomial time using an oracle deciding the original problem. Every NP-complete problem is self-reducible. It is conjectured that the integer factorization problem is not self-reducible.
Reductions and complete problems
Function problems can be reduced much like decision problems: Given function problems and we say that reduces to if there exists polynomially-time computable functions and such that for all instances and it holds that
It is therefore possible to define FNP-complete problems analogous to the NP-complete problem:
A problem is FNP-complete if every problem in FNP can be reduced to . The complexity class of FNP-complete problems is denoted by FNP-C or FNPC. It coincides with . Hence the problem FSAT is also an FNP-complete problem, and it holds that if and only if .
Total function problems
The relation used to define function problems has the drawback of being incomplete: Not every input has a counterpart such that . Therefore the question of computability of proofs 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. It has been shown that . In addition, if TFNP contains any FNP-complete problem it follows that .
- Raymond Greenlaw, H. James Hoover, Fundamentals of the theory of computation: principles and practice, Morgan Kaufmann, 1998, ISBN 1-55860-474-X, p. 45-51
- Elaine Rich, Automata, computability and complexity: theory and applications, Prentice Hall, 2008, ISBN 0-13-228806-0, section 28.10 "The problem classes FP and FNP", pp. 689-694