{{Short description|Notion of the "hardest" or "most general" problem in a complexity class}} {{Refimprove|date=October 2008}} In computational complexity theory, a computational problem is '''complete''' for a complexity class 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, 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.

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 computable enumeration have known complete problems, whereas classes that lack a computable 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<sup>''M''</sup> (BPP with oracle ''M'') has no complete problems.<ref>{{Cite book | doi=10.1007/BFb0012797| chapter=On relativization and the existence of complete sets| title=Automata, Languages and Programming| volume=140| pages=523–531| series=Lecture Notes in Computer Science| year=1982| last1=Sipser| first1=Michael| isbn=978-3-540-11576-2}}</ref>

== References == {{reflist}}

Category:Computational complexity theory