• Source: NP/poly

In computational complexity theory, NP/poly is a complexity class, a non-uniform analogue of the class NP of problems solvable in polynomial time by a non-deterministic Turing machine. It is the non-deterministic complexity class corresponding to the deterministic class P/poly.




Definition


NP/poly is defined as the class of problems solvable in polynomial time by a non-deterministic Turing machine that has access to a polynomial-bounded advice function.
It may equivalently be defined as the class of problems such that, for each instance size



n


{\displaystyle n}

, there is a Boolean circuit of size polynomial in



n


{\displaystyle n}

that implements a verifier for the problem. That is, the circuit computes a function



f
(
x
,
y
)


{\displaystyle f(x,y)}

such that an input



x


{\displaystyle x}

of length



n


{\displaystyle n}

is a yes-instance for the problem if and only if there exists



y


{\displaystyle y}

for which



f
(
x
,
y
)


{\displaystyle f(x,y)}

is true.




Applications


NP/poly is used in a variation of Mahaney's theorem on the non-existence of sparse NP-complete languages. Mahaney's theorem itself states that the number of yes-instances of length



n


{\displaystyle n}

of an NP-complete problem cannot be polynomially bounded unless P = NP. According to the variation, the number of yes-instances must be at least




2


n

ϵ






{\displaystyle 2^{n^{\epsilon }}}

for some



ϵ
>
0


{\displaystyle \epsilon >0}

and for infinitely many



n


{\displaystyle n}

, unless co-NP is a subset of NP/poly, which (by the Karp–Lipton theorem) would cause the collapse of the polynomial hierarchy.
The same computational hardness assumption that co-NP is not a subset of NP/poly also implies several other results in complexity such as the optimality of certain kernelization techniques.




References

Kata Kunci Pencarian:

• Litium
• Masalah pembenaman Connes
• Timbal(II) nitrat
• Metaloid
• Prokrastinasi
• Gastrin
• Germanium
• Neodimium(II) iodida
• NP/poly
• P/poly
• Kernelization
• Karp–Lipton theorem
• Steiner tree problem
• NP (complexity)
• Arthur–Merlin protocol
• Co-NP
• Probabilistically checkable proof
• Advice (complexity)