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

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 $n$ 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 $2^{n^{\epsilon }}$ for some $\epsilon >0$ and for infinitely many $n$ , 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]

References

Arora, Sanjeev; Barak, Boaz (2009), "Exercise 7.7", Computational Complexity: A Modern Approach, Cambridge University Press, p. 141, ISBN 9781139477369
Buhrman, Harry; Hitchcock, John M. (2008), "NP-hard sets are exponentially dense unless coNP ⊆ NP/poly", Twenty-Third Annual IEEE Conference on Computational Complexity, Los Alamitos, California: IEEE Computer Society, pp. 1–7, doi:10.1109/CCC.2008.21, MR 2513482, S2CID 2664381
Dell, Holger; van Melkebeek, Dieter (2014), "Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses", Journal of the ACM, 61 (4): A23:1–A23:27, doi:10.1145/2629620, MR 3250069, S2CID 1635025

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[1] Arora, Sanjeev; Barak, Boaz (2009), "Exercise 7.7", Computational Complexity: A Modern Approach, Cambridge University Press, p. 141, ISBN 9781139477369

[2] Buhrman, Harry; Hitchcock, John M. (2008), "NP-hard sets are exponentially dense unless coNP ⊆ NP/poly", Twenty-Third Annual IEEE Conference on Computational Complexity, Los Alamitos, California: IEEE Computer Society, pp. 1–7, doi:10.1109/CCC.2008.21, MR 2513482, S2CID 2664381

[3] Dell, Holger; van Melkebeek, Dieter (2014), "Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses", Journal of the ACM, 61 (4): A23:1–A23:27, doi:10.1145/2629620, MR 3250069, S2CID 1635025