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Concept# Search problem

Summary

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
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.
A search problem is often characterized by:
A set of states
A start state
A goal state or goal test: a boolean function which tells us whether a given state is a goal state
A successor function: a mapping from a state to a set of new states
Find a solution when not given an algorithm to solve a problem, but only a specification of what a solution looks like.

Official source

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Related publications (3)

Related concepts (9)

Ontological neighbourhood

In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring "Given a positive integer n, find a nontrivial prime factor of n." is a computational problem. A computational problem can be viewed as a set of instances or cases together with a, possibly empty, set of solutions for every instance/case. For example, in the factoring problem, the instances are the integers n, and solutions are prime numbers p that are the nontrivial prime factors of n.

Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is closely linked to the existence of an algorithm to solve the problem. The most widely studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power.

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 functional problem is defined by a relation over strings of an arbitrary alphabet : An algorithm solves if for every input such that there exists a satisfying , the algorithm produces one such , and if there are no such , it rejects.

This paper addresses the problem of transforming arbitrary data into binary data. This is intended as preprocessing for a supervised classification task. As a binary mapping compresses the total information of the dataset, the goal here is to design such a ...

We study the search problem class PPA_q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA_p for prime p. ...

This paper addresses the problem of transforming arbitrary data into binary data. This is intended as preprocessing for a supervised classification task. As a binary mapping compresses the total information of the dataset, the goal here is to design such a ...