In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that they are the ones most likely not to be in P. 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 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.
Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then P = NP, a critical foundation for Mahaney's theorem.
A decision problem 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. 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.
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