co-NP

Unsolved problem in computer science:

Is NP = co-NP ?
(more unsolved problems in computer science)

In computational complexity theory, co-NP is a complexity class. A decision problem ${\mathcal{X}}$ is a member of co-NP if and only if its complement $\overline{\mathcal{X}}$ is in the complexity class NP. Instances of decision problems in co-NP are sometimes called "counterexamples". In simple terms, co-NP is the class of problems for which efficiently verifiable proofs of "no" instances exist. Equivalently, co-NP is the set of decision problems where the "no" instances can be solved in polynomial time by a theoretical non-deterministic Turing machine.

An example of an NP-complete problem is the subset sum problem: given a finite set of integers, is there a non-empty subset that sums to zero? To give a proof of a "yes" instance, one must specify a non-empty subset that does sum to zero. The complementary problem is in co-NP and asks: "given a finite set of integers, does every non-empty subset have a non-zero sum?".

Relationship to other classes

P, the class of polynomial time solvable problems, is a subset of both NP and co-NP. P is thought to be a strict subset in both cases (and demonstrably cannot be strict in one case and not strict in the other). NP and co-NP are also thought to be unequal.^[1] If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP.^[2]

This can be shown as follows. Suppose there exists an NP-complete problem ${\mathcal{X}}$ that is in co-NP. Since all problems in NP can be reduced to ${\mathcal{X}}$ , it follows that for every problem in NP we can construct a non-deterministic Turing machine that decides its complement in polynomial time, i.e., NP ⊆ co-NP. From this it follows that the set of complements of the problems in NP is a subset of the set of complements of the problems in co-NP, i.e., co-NP ⊆ NP. Thus co-NP = NP. The proof that no co-NP-complete problem can be in NP if NP ≠ co-NP is symmetrical.

If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete (since otherwise NP = co-NP).

Integer factorization is closely related to the primality problem. An integer factorization algorithm can decide primality. The contrary is not true: for a primality test it is enough to show that a factor exists when checking a composite number. Both primality testing and factorization have long been known to be NP and co-NP problems. The AKS primality test, published in 2002, proves that primality testing also lies in P.^[3] Factorization may or may not have a polynomial-time algorithm.

References

↑ Hopcroft, John E. (2000). Introduction to Automata Theory, Languages, and Computation (2nd Edition). Boston: Addison-Wesley. ISBN 0-201-44124-1. Chap. 11.
↑ Goldreich, Oded (2010). P, NP, and NP-completeness: The Basics of Computational Complexity. Cambridge University Press. p. 155. ISBN 9781139490092.
↑ Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781-793.

External links

Complexity Zoo: coNP

Important complexity classes (more)

Considered feasible	DLOGTIME AC⁰ ACC⁰ TC⁰ L SL RL NL NC SC CC P P-complete ZPP RP BPP BQP APX

Suspected infeasible	UP NP NP-complete NP-hard co-NP co-NP-complete AM QMA PH ⊕P PP #P #P-complete IP PSPACE PSPACE-complete

Considered infeasible	EXPTIME NEXPTIME EXPSPACE ELEMENTARY PR R RE ALL

Class hierarchies	Polynomial hierarchy Exponential hierarchy Grzegorczyk hierarchy Arithmetical hierarchy Boolean hierarchy

Families of classes	DTIME NTIME DSPACE NSPACE Probabilistically checkable proof Interactive proof system

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