In complexity theory, the complexity class Co-NP-complete is the set of problems that are the hardest problems in Co-NP, in the sense that they are the ones most likely not to be in P. If you can find a way to solve a Co-NP-complete problem quickly, then you can use that algorithm to solve all Co-NP problems quickly.

A more formal definition: A decision problem C is Co-NP-complete if it is in Co-NP and if every problem in Co-NP is 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. As a consequence, if we had a polynomial time algorithm for C, we could solve all Co-NP problems in polynomial time.

One simple 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, which asks whether there exists at least one such assignment.

Each Co-NP-Complete problem is the complement of an NP-complete problem. The two sets are either equal or disjoint. The latter is thought more likely, but this is not known. See Co-NP and NP-complete for more details.