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 formula_1, there is a Boolean circuit of size polynomial in formula_1 that implements a verifier for the problem. That is, the circuit computes a function formula_3 such that an input formula_4 of length formula_1 is a yes-instance for the problem if and only if there exists formula_6 for which formula_3 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 formula_1 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 formula_9 for some formula_10 and for infinitely many formula_1, 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.
 ;templatestyles src="Reflist/styles.css" / ;