computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...

, a

computational problem 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 probl ...

is complete for a

complexity class In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms o ...

if it is, in a technical sense, among the "hardest" (or "most expressive") problems in the complexity class. More formally, a problem ''p'' is called hard for a complexity class ''C'' under a given type of reduction if there exists a reduction (of the given type) from any problem in ''C'' to ''p''. If a problem is both hard for the class and a member of the class, it is complete for that class (for that type of reduction). A problem that is complete for a class ''C'' is said to be C-complete, and the class of all problems complete for ''C'' is denoted C-complete. The first complete class to be defined and the most well known is

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...

, a class that contains many difficult-to-solve problems that arise in practice. Similarly, a problem hard for a class ''C'' is called C-hard, e.g.

NP-hard In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...

. Normally, it is assumed that the reduction in question does not have higher computational complexity than the class itself. Therefore, it may be said that if a ''C-complete'' problem has a "computationally easy" solution, then all problems in "C" have an "easy" solution. Generally, complexity classes that have a recursive enumeration have known complete problems, whereas classes that lack a recursive enumeration have none. For example, NP, co-NP, PLS, PPA all have known natural complete problems. There are classes without complete problems. For example, Sipser showed that there is a language M such that BPP^M (BPP with

oracle An oracle is a person or agency considered to provide wise and insightful counsel or prophetic predictions, most notably including precognition of the future, inspired by deities. As such, it is a form of divination. Description The word ...

M) has no complete problems.

References

{{reflist Computational complexity theory