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| {{Correct title|title=#P|reason=hash}}
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| In [[computational complexity theory]], the complexity class '''#P''' (pronounced "number P" or, sometimes "sharp P" or "hashtag P") is the set of the counting problems associated with the [[decision problem]]s in the set '''[[NP (complexity)|NP]]'''. More formally, '''#P''' is the class of function problems of the form "compute ''ƒ''(''x'')", where ''ƒ'' is the number of accepting paths of a [[nondeterministic Turing machine]] running in polynomial time. Unlike most well-known complexity classes, it is not a class of [[decision problem]]s but a class of [[function problem]]s.
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| An '''NP''' problem is often of the form "Are there any solutions that satisfy certain constraints?" For example:
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| * Are there any subsets of a list of integers that add up to zero? ([[subset sum problem]])
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| * Are there any [[Hamiltonian cycle]]s in a given [[graph theory|graph]] with cost less than 100? ([[traveling salesman problem]])
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| * Are there any variable assignments that satisfy a given [[conjunctive normal form|CNF]] formula? ([[Boolean satisfiability problem]])
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| The corresponding '''#P''' problems ask "how many" rather than "are there any". For example:
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| * How many subsets of a list of integers add up to zero?
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| * How many Hamiltonian cycles in a given graph have cost less than 100?
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| * How many variable assignments satisfy a given CNF formula?
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| Clearly, a '''#P''' problem must be at least as hard as the corresponding '''NP''' problem. If it's easy to count answers, then it must be easy to tell whether there are any answers – just count them and see whether the count is greater than zero.
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| One consequence of [[Toda's theorem]] is that a polynomial-time machine with a '''#P''' [[oracle machine|oracle]] ('''P'''<sup>'''#P'''</sup>) can solve all problems in '''[[PH (complexity)|PH]]''', the entire [[polynomial hierarchy]]. In fact, the polynomial-time machine only needs to make one '''#P''' query to solve any problem in '''PH'''. This is an indication of the extreme difficulty of solving '''#P'''-complete problems exactly.
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| Surprisingly, some '''#P''' problems that are believed to be difficult correspond to easy '''[[P (complexity)|P]]''' problems. For more information on this, see [[sharp-P-complete|#P-complete]].
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| The closest decision problem class to '''#P''' is '''[[PP (complexity)|PP]]''', which asks whether a majority (more than half) of the computation paths accept. This finds the most significant bit in the '''#P''' problem answer. The decision problem class '''[[Parity P|⊕P]]''' instead asks for the least significant bit of the '''#P''' answer.
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| The complexity class '''#P''' was first defined by [[Leslie Valiant]] in a 1979 article on the computation of the [[permanent]], in which he proved that [[permanent is sharp-P-complete|permanent is #P-complete]].<ref>{{cite journal
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| | author = Leslie G. Valiant
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| | title = The Complexity of Computing the Permanent
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| | journal = Theoretical Computer Science
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| | volume = 8
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| | pages = 189–201
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| | publisher = [[Elsevier]]
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| | location =
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| | date = 1979
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| | doi = 10.1016/0304-3975(79)90044-6
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| | issue = 2}}</ref>
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| [[Larry Stockmeyer]] has proved that for every #P problem ''P'' there exists a randomized algorithm using oracle for SAT, which given an instance ''a'' of ''P'' and ''ε'' > 0 returns with high probability a number ''x'' such that <math>(1-\epsilon) P(a) \leq x \leq (1+\epsilon) P(a)</math>. The runtime of the algorithm is polynomial in ''a'' and 1/''ε''. The algorithm is based on [[leftover hash lemma]]. | |
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| == References ==
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| <references/>
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| == External links ==
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| * {{CZoo|Class #P|Symbols#sharpp}}
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| {{ComplexityClasses}}
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| [[Category:Complexity classes]]
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