# NP-intermediate

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: Schaefer's dichotomy theorem provides conditions under which classes of constrained Boolean satisfiability problems can not be in NPI.[2] Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm.[3]

## References

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. ^ Grädel, Erich; Kolaitis, Phokion G.; Libkin, Leonid; Maarten, Marx; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (2007). Finite model theory and its applications. Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer-Verlag. p. 348. ISBN 978-3-540-00428-8. Zbl 1133.03001.
3. ^ "Problems Between P and NPC". Theoretical Computer Science Stack Exchange. 20 August 2011. Retrieved 1 November 2013.
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14. ^ 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
15. ^ http://cstheory.stackexchange.com/questions/3826/np-hardness-of-a-special-case-of-orthogonal-packing-problem/3827#3827
16. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/3950#3950
17. ^ Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
18. ^ http://www.openproblemgarden.org/?q=op/theoretical_computer_science/subset_sums_equality
19. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4010#4010
20. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/6384#6384
21. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/7806#7806
22. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4331#4331