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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 theoryEdit
- 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 topologyEdit
- 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
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- Rotation distance, triangulations, and hyperbolic geometry
- Reconstructing sets from interpoint distances
- Demaine, Erik D.; O'Rourke, Joseph (2007), "24 Geodesics: Lyusternik–Schnirelmann", Geometric folding algorithms: Linkages, origami, polyhedra, Cambridge: Cambridge University Press, pp. 372–375, doi:10.1017/CBO9780511735172, ISBN 978-0-521-71522-5, MR 2354878.
- Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
- 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.
- Nishimura, N.; Ragde, P.; Thilikos, D.M. (2002), "On graph powers for leaf-labeled trees", Journal of Algorithms, 42: 69–108, doi:10.1006/jagm.2001.1195.
- Fellows, Michael R.; Rosamond, Frances A.; Rotics, Udi; Szeider, Stefan (2009), "Clique-width is NP-complete", SIAM Journal on Discrete Mathematics, 23 (2): 909–939, doi:10.1137/070687256, MR 2519936.
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- 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.