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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 E. 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, it follows 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 cannot 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.
Algebra and number theory
- Factoring integers
- Discrete Log Problem and others related to cryptographic assumptions
- Isomorphism problems: Group isomorphism problem, Group automorphism, Ring isomorphism, Ring automorphism
- Numbers in boxes problems
- The linear divisibility problem
Computational geometry and computational topology
- 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
- The cutting stock problem with a constant number of object lengths
- Knot triviality
- Deciding whether a given triangulated 3-manifold is a 3-sphere
- Gap version of the closest vector in lattice problem
- Finding a simple closed quasigeodesic on a convex polyhedron
- Determining winner in parity games
- Determining who has the highest chance of winning a stochastic game
- Agenda control for balanced single-elimination tournaments
- Graph isomorphism problem
- Planar minimum bisection
- Deciding whether a graph admits a graceful labeling
- Clustered planarity
- Recognizing leaf powers and k-leaf powers
- Recognizing graphs of bounded clique-width
- Finding a simultaneous embedding with fixed edges
- Assuming NEXP is not equal to EXP, padded versions of NEXP-complete problems
- Problems in TFNP
- Pigeonhole subset sum
- Finding the VC dimension
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- Rotation distance, triangulations, and hyperbolic geometry
- Reconstructing sets from interpoint distances
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- Complexity Zoo: Class NPI
- Basic structure, Turing reducibility and NP-hardness
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