NP/poly

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 ${\displaystyle n}$, there is a Boolean circuit of size polynomial in ${\displaystyle n}$ that implements a verifier for the problem. That is, the circuit computes a function ${\displaystyle f(x,y)}$ such that an input ${\displaystyle x}$ of length ${\displaystyle n}$ is a yes-instance for the problem if and only if there exists ${\displaystyle y}$ for which ${\displaystyle f(x,y)}$ is true.^[1]

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 ${\displaystyle n}$ 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 ${\displaystyle 2^{n^{\epsilon }}}$ for some ${\displaystyle \epsilon >0}$ and for infinitely many ${\displaystyle n}$, unless co-NP is a subset of NP/poly, which (by the Karp–Lipton theorem) would cause the collapse of the polynomial hierarchy.^[2] 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.^[3]