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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 recursive enumeration have known complete problems, whereas those that do not, don't have any known complete problems. For example, NP, co-NP, PLS, PPA all have known natural complete problems, while RP, ZPP, BPP and TFNP do not have any known complete problems (although such a problem may be discovered in the future). There are classes without complete problems. For example, Sipser showed that there is a language M such that BPPM (BPP with oracle M) has no complete problems.〔M. Sipser. On relativization and the existence of complete sets, Proceedings of ICALP'82, Springer-Verlag Lecture Notes in Computer Science volume 140, pp. 523-531, 1982.〕 == References == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complete (complexity)」の詳細全文を読む スポンサード リンク
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