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In
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, a function problem is a
computational problem In theoretical computer science, a computational problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computati ...
where a single output (of a
total function In mathematics, a partial function from a set to a set is a function from a subset of (possibly the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain o ...
) is expected for every input, but the output is more complex than that of a
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
. For function problems, the output is not simply 'yes' or 'no'.


Definition

A functional problem P is defined by a relation R over strings of an arbitrary
alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...
\Sigma: : R \subseteq \Sigma^* \times \Sigma^*. An algorithm solves P if for every input x such that there exists a y satisfying (x, y) \in R, the algorithm produces one such y, and if there are no such y, it rejects. A promise function problem is allowed to do anything (thus may not terminate) if no such y exists.


Examples

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 boolean formula \varphi with variables x_1, \ldots, x_n, find an assignment x_i \rightarrow \ such that \varphi evaluates to \text or decide that no such assignment exists. In this case the relation R is given by tuples of suitably encoded boolean formulas and satisfying assignments. While a SAT algorithm, fed with a formula \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 In the Computational complexity theory, theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible ...
, which asks for the route taken by the salesman, and the integer factorization problem, which asks for the list of factors.


Relationship to other complexity classes

Consider an arbitrary
decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
L in the class NP. By the definition of NP, each problem instance x that is answered 'yes' has a polynomial-size certificate y which serves as a proof for the 'yes' answer. Thus, the set of these tuples (x,y) forms a relation, representing the function problem "given x in L, find a certificate y for x". This function problem is called the ''function variant'' of L; it belongs to the class FNP. 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 whose solutions can be found in polynomial time.


Self-reducibility

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 \varphi is satisfiable. After that the algorithm can fix variable x_1 to TRUE and ask again. If the resulting formula is still satisfiable the algorithm keeps x_1 fixed to TRUE and continues to fix x_2, otherwise it decides that x_1 has to be FALSE and continues. Thus, FSAT is solvable in polynomial time using an
oracle An oracle is a person or thing considered to provide insight, wise counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. If done through occultic means, it is a form of divination. Descript ...
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 In computational complexity theory, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ...
problem is self-reducible. It is conjectured that the integer factorization problem is not self-reducible, because deciding whether an integer is prime is in P (easy), while the integer factorization problem is believed to be hard for a classical computer. There are several (slightly different) notions of self-reducibility.


Reductions and complete problems

Function problems can be reduced much like decision problems: Given function problems \Pi_R and \Pi_S we say that \Pi_R reduces to \Pi_S if there exists polynomially-time computable functions f and g such that for all instances x of R and possible solutions y of S, it holds that *If x has an R-solution, then f(x) has an S-solution. *(f(x), y) \in S \implies (x, g(x,y)) \in R. It is therefore possible to define FNP-complete problems analogous to the NP-complete problem: A problem \Pi_R is FNP-complete if every problem in FNP can be reduced to \Pi_R. The complexity class of FNP-complete problems is denoted by FNP-C or FNPC. Hence the problem FSAT is also an FNP-complete problem, and it holds that \mathbf = \mathbf if and only if \mathbf = \mathbf.


Total function problems

The relation R(x, y) used to define function problems has the drawback of being incomplete: Not every input x has a counterpart y such that (x, y) \in R. 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 game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...
in certain strategic games where a solution is guaranteed to exist. In addition, if TFNP contains any FNP-complete problem it follows that \mathbf = \textbf.


See also

*
Decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natura ...
*
Search problem In computational complexity theory and computability theory, a search problem is a computational problem of finding an ''admissible'' answer for a given input value, provided that such an answer exists. In fact, a search problem is specified by a b ...
*
Counting problem (complexity) In computational complexity theory and computability theory, a counting problem is a type of computational problem. If ''R'' is a search problem then :c_R(x)=\vert\\vert \, is the corresponding counting function and :\#R=\ denotes the corre ...
*
Optimization problem In mathematics, engineering, computer science and economics Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goo ...


References

* Raymond Greenlaw, H. James Hoover, ''Fundamentals of the theory of computation: principles and practice'', Morgan Kaufmann, 1998, , p. 45-51 * Elaine Rich, ''Automata, computability and complexity: theory and applications'', Prentice Hall, 2008, , section 28.10 "The problem classes FP and FNP", pp. 689–694 {{refend Computational problems Functions and mappings