Counting problem (complexity)#Counting complexity class

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

: $c_R(x)=\vert\{y\mid R(x,y)\}\vert \,$

is the corresponding counting function and

: $\#R=\{(x,y)\mid y\leq c_R(x)\}$

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.{{cite web |last1=Barak |first1=Boaz |title=Complexity of counting |url=https://www.cs.princeton.edu/courses/archive/spring06/cos522/count.pdf |website=Princeton University |date=Spring 2006}}

== See also ==

GapP

References

External links

{{PlanetMath reference|urlname=CountingProblem|title=counting problem}}
{{PlanetMath reference|urlname=CountingComplexityClass|title=counting complexity class}}

Category:Computational problems