Counting problem (complexity)
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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
is the corresponding counting function and
denotes the corresponding decision problem.
Note that cR is a search problem while #R is a decision problem, however cR can be C
binary search
(the reason #R is defined the way it is, rather than being the graph of cR, is to make this binary search possible).
Counting complexity class
If NX is a complexity class associated with
#P is the class of counting problems associated with NP
search problems.
Just as NP has NP-complete problems via many-one reductions, #P has complete problems via parsimonious reductions
, problem transformations that preserve the number of solutions.