Computational problems

In theoretical computer science, a computational problem is a mathematical object representing a collection of questions that computers might be able to solve. For example, the problem of factoring

is a computational problem. Computational problems are one of the main objects of study in theoretical computer science. The field of algorithms studies methods of solving computational problems efficiently. The complementary field of computational complexity attempts to explain why certain computational problems are intractable for computers.

A computational problem can be viewed as an infinite collection of instances together with a solution for every instance. For example, in the factoring problem, the instances are the integers n, and solutions are prime numbers p that describe nontrivial prime factors of n.

It is conventional to represent both instances and solutions by binary strings, namely elements of {0, 1}^*. For example, numbers can be represented as binary strings using the binary encoding. (For readability, we identify numbers with their binary encodings in the examples below.)

A decision problem is a computational problem where the answer for every instance is either yes or no. An example of a decision problem is primality testing:

A decision problem is typically represented as the set of all instances for which the answer is yes. For example, primality testing can be represented as the infinite set

In a search problem, the answers can be arbitrary strings. For example, factoring is a search problem where the instances are (string representations of) positive integers and the solutions are (string representations of) collections of primes.

A search problem is represented as a relation consisting of all the instance-solution pairs, called a search relation. For example, factoring can be represented as the relation

which consist of all pairs of numbers (n, p), where p is a nontrivial prime factor of n.

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