In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-clas ...

of an

NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ...

problem. There are some problems in both NP and co-NP, for example all problems in P or

integer factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...

. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and

for more details. Fortune showed in 1979 that if any

sparse language In computational complexity theory, a sparse language is a formal language (a set of strings) such that the complexity function, counting the number of strings of length ''n'' in the language, is bounded by a polynomial function of ''n''. They a ...

is co-NP-complete (or even just co-NP-hard), then , a critical foundation for

Mahaney's theorem Mahaney's theorem is a theorem in computational complexity theory proven by Stephen Mahaney that states that if any sparse language is NP-complete, then P = NP. Also, if any sparse language is NP-complete with respect to Turing reduction In comput ...

Formal definition

decision problem In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm whe ...

''C'' is co-NP-complete if it is in co-NP and if every problem in co-NP is polynomial-time many-one reducible to it. This means that for every co-NP problem ''L'', there exists a polynomial time algorithm which can transform any instance of ''L'' into an instance of ''C'' with the same

truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pro ...

. As a consequence, if we had a polynomial time algorithm for ''C'', we could solve all co-NP problems in polynomial time.

Example

One example of a co-NP-complete problem is tautology, the problem of determining whether a given Boolean formula is a tautology; that is, whether every possible assignment of true/false values to variables yields a true statement. This is closely related to the

Boolean satisfiability problem In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisf ...

, which asks whether there exists ''at least one'' such assignment, and is NP-complete.

References

External links

* {{ComplexityClasses Complexity classes