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In computational complexity theory and computability theory, a counting problem is a type of computational problem. If R is a search problem then

${\displaystyle c_{R}(x)=\vert \{y\mid R(x,y)\}\vert \,}$

is the corresponding counting function and

${\displaystyle \#R=\{(x,y)\mid y\leq c_{R}(x)\}}$

denotes the corresponding decision problem.

Note that c_R is a search problem while #R is a decision problem, however c_R can be C Cook-reduced to #R (for appropriate C) using a binary search (the reason #R is defined the way it is, rather than being the graph of c_R, is to make this binary search possible).

Counting complexity class

Just as NP has NP-complete problems via many-one reductions, #P has #P-complete problems via parsimonious reductions, problem transformations that preserve the number of solutions.

See also

• GapP

References

1. Barak, Boaz (Spring 2006). "Complexity of counting" (PDF). Princeton University.

External links

• "counting problem". PlanetMath.
• "counting complexity class". PlanetMath.

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