Counting problem (complexity)

In computational complexity theory and computability theory, a counting problem is a type of computational problem. If R is a search problem (formalised as a binary relation) then

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

is the corresponding counting function and

$\mathrm{\#}R=\{(x,y)\mid y\le c_R(x)\}$

denotes the corresponding decision problem.

Note that c_R is a function 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

See also: #P and #P-complete

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