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 $n$, 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.{{citation|title=Computational Complexity: A Modern Approach|first1= Sanjeev|last1=Arora|first2=Boaz|last2=Barak|publisher=Cambridge University Press|year=2009|isbn=9781139477369|contribution=Exercise 7.7|contribution-url=https://books.google.com/books?id=nGvI7cOuOOQC&pg=PA141|page=141}}

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 $2^\{n^\backslash epsilon\}$ for some $\backslash epsilon>0$ 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.{{citation

| last1 = Buhrman | first1 = Harry

| last2 = Hitchcock | first2 = John M.

| contribution = NP-hard sets are exponentially dense unless coNP ⊆ NP/poly

| doi = 10.1109/CCC.2008.21

| mr = 2513482

| pages = 1–7

| publisher = IEEE Computer Society | location = Los Alamitos, California

| title = Twenty-Third Annual IEEE Conference on Computational Complexity

| year = 2008| s2cid = 2664381

| url = https://ir.cwi.nl/pub/13767

}}

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.{{citation

| last1 = Dell | first1 = Holger

| last2 = van Melkebeek | first2 = Dieter

| doi = 10.1145/2629620

| issue = 4

| journal = Journal of the ACM

| mr = 3250069

| page = A23:1–A23:27

| title = Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses

| volume = 61

| year = 2014| s2cid = 1635025

| url = https://drops.dagstuhl.de/opus/volltexte/2010/2504/

}}

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Category:Complexity classes