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<title>Complete (complexity)</title>

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<p>In <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, a <a href="Computational_problem" title="Computational problem">computational problem</a> is <b>complete</b> for a <a href="Complexity_class" title="Complexity class">complexity class</a> if it is, in a technical sense, among the "hardest" (or "most expressive") problems in the complexity class.

</p><p>More formally, a problem <i>p</i> is called <b>hard</b> for a complexity class <i>C</i> under a given type of <a href="Reduction_(complexity)" title="Reduction (complexity)">reduction</a> if there exists a reduction (of the given type) from any problem in <i>C</i> to <i>p</i>. If a problem is both <b>hard</b> for the class and a member of the class, it is <b>complete</b> for that class (for that type of reduction).

</p><p>A problem that is complete for a class <i>C</i> is said to be <b>C-complete</b>, and the class of all problems complete for <i>C</i> is denoted <b>C-complete</b>. The first complete class to be defined and the most well known is <a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a>, a class that contains many difficult-to-solve problems that arise in practice. Similarly, a problem hard for a class <i>C</i> is called <b>C-hard</b>, e.g. <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>.

</p><p>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 <i>C-complete</i> problem has a "computationally easy" solution, then all problems in "C" have an "easy" solution.

</p><p>Generally, complexity classes that have a recursive enumeration have known complete problems, whereas classes that lack a recursive enumeration have none. For example, <a href="NP_(complexity)" title="NP (complexity)">NP</a>, <a href="Co-NP" title="Co-NP">co-NP</a>, <a href="PLS_(complexity)" title="PLS (complexity)">PLS</a>, <a href="PPA_(complexity)" title="PPA (complexity)">PPA</a> all have known natural complete problems.

</p><p>There are classes without complete problems. For example, Sipser showed that there is a language <b>M</b> such that <b>BPP</b><sup><b>M</b></sup> (<b>BPP</b> with <a href="Oracle_machine" title="Oracle machine">oracle</a> <b>M</b>) has no complete problems.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFSipser1982" class="citation book cs1">Sipser, Michael (1982). "On relativization and the existence of complete sets". <i>Automata, Languages and Programming</i>. Lecture Notes in Computer Science. Vol. 140. pp. <span class="nowrap">523–</span>531. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0012797">10.1007/BFb0012797</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-11576-2</bdi>.</cite></span>

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