Search problem

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In the mathematics of computational complexity theory, computability theory, and decision theory, a search problem is a type of computational problem represented by a binary relation. Intuitively, the problem consists in finding structure "y" in object "x". An algorithm is said to solve the problem if at least one corresponding structure exists, and then one occurrence of this structure is made output; otherwise, the algorithm stops with an appropriate output ("not found" or any message of the like).

Every search problem also has a corresponding decision problem, namely

${\displaystyle L(R)=\{x\mid \exists yR(x,y)\}.\,}$

This definition may be generalized to n-ary relations using any suitable encoding which allows multiple strings to be compressed into one string (for instance by listing them consecutively with a delimiter).

More formally, a relation R can be viewed as a search problem, and a Turing machine which calculates R is also said to solve it. More formally, if R is a binary relation such that field(R) ⊆ Γ⁺ and T is a Turing machine, then T calculates R if:

• If x is such that there is some y such that R(x, y) then T accepts x with output z such that R(x, z) (there may be multiple y, and T need only find one of them)
• If x is such that there is no y such that R(x, y) then T rejects x

(Note that the graph of a partial function is a binary relation, and if T calculates a partial function then there is at most one possible output.)

Such problems occur very frequently in graph theory and combinatorial optimization, for example, where searching for structures such as particular matchings, optional cliques, particular stable sets, etc. are subjects of interest.