Search problem

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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 binary relation R where xRy if and only if "y is an admissible answer given x".^{[note 1]} Search problems frequently occur in graph theory and combinatorial optimization, e.g. searching for matchings, optional cliques, and stable sets in a given undirected graph.

An algorithm is said to solve a search problem if, for every input value x, it returns an admissible answer y for x when such an answer exists; otherwise, it returns any appropriate output, e.g. "not found" for x with no such answer.

PlanetMath defines the problem as follows:^[1]

If ${\displaystyle R}$ is a binary relation such that ${\displaystyle \operatorname {field} (R)\subseteq \Gamma ^{+}}$ and ${\displaystyle T}$ is a Turing machine, then ${\displaystyle T}$ calculates ${\displaystyle f}$ if:^{[note 2]}

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

Note that the graph of a partial function is a binary relation, and if ${\displaystyle T}$ calculates a partial function then there is at most one possible output.

A ${\displaystyle R}$ can be viewed as a search problem, and a Turing machine which calculates ${\displaystyle R}$ is also said to solve it. Every search problem has a corresponding decision problem, namely ${\displaystyle L(R)=\{x\mid \exists yR(x,y)\}.}$

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

• Unbounded search operator
• Decision problem
• Optimization problem
• Counting problem (complexity)
• Function problem
• Search games

This article incorporates material from search problem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.