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#P-complete, pronounced "sharp P complete" or "number P complete" is a complexity class in computational complexity theory. A problem is #P-complete if and only if it is in #P, and every problem in #P can be reduced to it by a polynomial-time counting reduction, i.e. a polynomial-time Turing reduction relating the cardinalities of solution sets.[citation needed] Equivalently, a problem is #P-complete if and only if it is in #P, and for any non-deterministic Turing machine ("NP machine"), the problem of computing its number of accepting paths can be reduced to this problem.[citation needed]

Very often the reductions are "parsimonious," i.e., they preserve the number of solutions.[citation needed]

Examples of #P-complete problems include:

A polynomial-time algorithm for solving a #P-complete problem, if it existed, would imply P = PH, and thus P = NP. No such algorithm is currently known.


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