Polynomial-time counting reduction explained
In the computational complexity theory of counting problems, a polynomial-time counting reduction is a type of reduction (a transformation from one problem to another) used to define the notion of completeness for the complexity class ♯P. These reductions may also be called polynomial many-one counting reductions or weakly parsimonious reductions; they are analogous to many-one reductions for decision problems and they generalize the parsimonious reductions.
Definition
A polynomial-time counting reduction is usually used to transform instances of a known-hard problem
into instances of another problem
that is to be proven hard. It consists of two functions
and
, both of which must be computable in polynomial time. The function
transforms inputs for
into inputs for
, and the function
transforms outputs for
into outputs for
.
These two functions must preserve the correctness of the output. That is, suppose that one transforms an input
for problem
to an input
for problem
, and then one solves
to produce an output
. It must be the case that the transformed output
is the correct output for the original input
. That is, if the input-output relations of
and
are expressed as functions, then their
function composition must obey the
identity
. Alternatively, expressed in terms of
algorithms, one possible algorithm for solving
would be to apply
to transform the problem into an instance of
, solve that instance, and then apply
to transform the output of
into the correct answer for
.
Relation to other kinds of reduction
As a special case, a parsimonious reduction is a polynomial-time transformation
on the inputs to problems that preserves the exact values of the outputs. Such a reduction can be viewed as a polynomial-time counting reduction, by using the
identity function as the function
.
Applications in complexity theory
A functional problem (specified by its inputs and desired outputs) belongs to the complexity class ♯P if there exists a non-deterministic Turing machine that runs in polynomial time, for which the output to the problem is the number of accepting paths of the Turing machine. Intuitively, such problems count the number of solutions to problems in the complexity class NP. A functional problem
is said to be ♯P-hard if there exists a polynomial-time counting reduction from every problem
in ♯P to
. If, in addition,
itself belongs to ♯P, then
is said to be
♯P-complete. (Sometimes, as in Valiant's original paper
proving the completeness of the permanent of 0–1 matrices, a weaker notion of reduction,
Turing reduction, is instead used for defining ♯P-completeness.)
The usual method of proving a problem
in ♯P to be ♯P-complete is to start with a single known ♯P-complete problem
and find a polynomial-time counting reduction from
to
. If this reduction exists, then there exists a reduction from any other problem in ♯P to
, obtained by
composing a reduction from the other problem to
with the reduction from
to
