# Co-NP

In computational complexity theory, co-NP is a complexity class, the complement of the complexity class NP. In simple terms, it is the class of problems for which efficiently verifiable proofs of no instances, sometimes called counterexamples, exist.

For example, there is an NP-complete problem called the subset-sum problem, which asks if any subset of a finite set of integers sums to zero. Its complement problem is in co-NP and asks if every subset sums to a nonzero number. A counterexample would be a subset which does sum to zero, which is easy to verify.

P is a subset of both NP and co-NP. That subset is thought to be strict in both cases. NP and co-NP are also thought to be unequal. If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP.

This can be shown as follows. Assume that there is an NP-complete problem that is in co-NP. Since all problems in NP can be reduced to this problem it follows that for all problems in NP we can construct a non-deterministic Turing machine that decides the complement of the problem in polynomial time, i.e., NP is a subset of 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 is a subset of NP. Since we already knew that NP is a subset of co-NP it follows that they are the same. The proof for the fact that no co-NP-complete problem can be in 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). One example is integer factorization, the problem of finding the prime factors of a number. It is in both NP and co-NP, but is generally suspected to be outside P, outside NP-complete, and outside co-NP-complete.

## References

• Complexity Zoo: coNP (http://www.complexityzoo.com/#conp)

 Important complexity classes (more) P | NP | Co-NP | NP-C | Co-NP-C | NP-hard | UP | #P | #P-C | L | NC | P-C PSPACE | PSPACE-C | EXPTIME | EXPSPACE | BQP | BPP | RP | ZPP | PCP | IP | PH
es:Co-NP

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