In computational complexity, problems that are in the complexity class NP but are neither in the class P nor NP-complete are called NP-intermediate, and the class of such problems is called NPI. Ladner's theorem, shown in 1975 by Richard Ladner, is a result asserting that, if P ≠ NP, then NPI is not empty; that is, NP contains problems that are neither in P nor NP-complete. Since the other direction is trivial, we can say that P = NP if and only if NPI is empty.
Under the assumption that P ≠ NP, Ladner explicitly constructs a problem in NPI, although this problem is artificial and otherwise uninteresting. It is an open question whether any "natural" problem has the same property. Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm.
Another problem in NP that is not known to be in P or NP-complete is the minimum circuit size problem (MCSP). Unlike the above problems, MCSP is believed to be NP-complete. However, proving as much via a many-one reduction will imply circuit lower bounds for E (unless NP is contained in SUBEXP, which is a violation of the exponential time hypothesis). Therefore, proving that MCSP is NP-complete is considered outside of current techniques.
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- Kabanets, Valentine; Cai, Jin-Yi (2000), "Circuit minimization problem", Proc. 32nd Symposium on Theory of Computing, Portland, Oregon, USA, pp. 73–79, doi:10.1145/335305.335314, ECCC TR99-045.
- Complexity Zoo: Class NPI
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