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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, 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. Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm.
- Factoring integers
- Isomorphism problems: Graph isomorphism problem, Group isomorphism problem, Group automorphism, Ring isomorphism, Ring automorphism
- Computing the rotation distance between two binary trees or the flip distance between two triangulations of the same convex polygon
- The turnpike problem of reconstructing points on line from their distance multiset
- Discrete Log Problem and others related to cryptographic assumptions
- Determining winner in parity games
- Determining who has the highest chance of winning a stochastic game
- Numbers in boxes problems
- Agenda control for balanced single-elimination tournaments
- Knot triviality
- Assuming NEXP is not equal to EXP, padded versions of NEXP-complete problems
- Problems in TFNP
- Intersecting Monotone SAT
- Minimum Circuit Size Problem
- Deciding whether a given triangulated 3-manifold is a 3-sphere
- The cutting stock problem with a constant number of object lengths
- Monotone self-duality
- Planar minimum bisection
- Pigeonhole subset sum
- Determining the result of a comparison between two sums of square roots of integers
- Deciding whether a graph admits a graceful labeling
- Gap version of the closest vector in lattice problem
- The linear divisibility problem
- Finding the VC dimension
- Clustered planarity
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- Rotation distance, triangulations, and hyperbolic geometry
- Reconstructing sets from interpoint distances
- On total functions, existence theorems and computational complexity
- 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
- Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
- Papadimitriou, Christos H.; Yannakakis, Mihalis (1996), "On limited nondeterminism and the complexity of the V-C dimension", Journal of Computer and System Sciences 53 (2, part 1): 161–170, doi:10.1006/jcss.1996.0058, MR 1418886
- Cortese, Pier Francesco; Di Battista, Giuseppe; Frati, Fabrizio; Patrignani, Maurizio; Pizzonia, Maurizio (2008), "C-planarity of C-connected clustered graphs", Journal of Graph Algorithms and Applications 12 (2): 225–262, doi:10.7155/jgaa.00165, MR 2448402.
- Complexity Zoo: Class NPI
- Basic structure, Turing reducibility and NP-hardness
- Lance Fortnow (24 March 2003). "Foundations of Complexity, Lesson 16: Ladner’s Theorem". Retrieved 1 November 2013.