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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,[1] 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.[2]

List of problems that might be NP-intermediate[3][edit]


  1. ^ Ladner, Richard (1975). "On the Structure of Polynomial Time Reducibility". Journal of the ACM (JACM) 22 (1): 155–171. doi:10.1145/321864.321877. 
  2. ^ "Problems Between P and NPC". Theoretical Computer Science Stack Exchange. 20 August 2011. Retrieved 1 November 2013. 
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  4. ^ Rotation distance, triangulations, and hyperbolic geometry
  5. ^ Reconstructing sets from interpoint distances
  6. ^ a b
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  10. ^ On total functions, existence theorems and computational complexity
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  13. ^ 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 
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  16. ^ Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
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