# 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 E. 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, 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 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]

## List of problems that might be NP-intermediate[4]

### Computational geometry and computational topology

• Computing the rotation distance[12] between two binary trees or the flip distance between two triangulations of the same convex polygon
• The turnpike problem[13] of reconstructing points on line from their distance multiset
• The cutting stock problem with a constant number of object lengths[14]
• Knot triviality[15]
• Deciding whether a given triangulated 3-manifold is a 3-sphere
• Gap version of the closest vector in lattice problem[16]
• Finding a simple closed quasigeodesic on a convex polyhedron[17]

### Game theory

• Determining winner in parity games[18]
• Determining who has the highest chance of winning a stochastic game[18]
• Agenda control for balanced single-elimination tournaments[19]

## 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; Marx, Maarten; 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.
4. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/237#237
5. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4010#4010
6. ^ http://blog.computationalcomplexity.org/2010/07/what-is-complexity-of-these-problems.html
7. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4331#4331
8. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1739#1739
9. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1745#1745
10. ^ 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
11. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/3950#3950
12. ^ Rotation distance, triangulations, and hyperbolic geometry
13. ^ Reconstructing sets from interpoint distances
14. ^ http://cstheory.stackexchange.com/questions/3826/np-hardness-of-a-special-case-of-orthogonal-packing-problem/3827#3827
15. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1106#1106
16. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/7806#7806
17. ^ 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.
18. ^ a b http://kintali.wordpress.com/2010/06/06/np-intersect-conp/
19. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/460#460
20. ^ Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
21. ^ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/6384#6384
22. ^ 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.
23. ^ 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.
24. ^ 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.
25. ^ Gassner, Elisabeth; Jünger, Michael; Percan, Merijam; Schaefer, Marcus; Schulz, Michael (2006), "Simultaneous graph embeddings with fixed edges", Graph-Theoretic Concepts in Computer Science: 32nd International Workshop, WG 2006, Bergen, Norway, June 22-24, 2006, Revised Papers, Lecture Notes in Computer Science, 4271, Berlin: Springer, pp. 325–335, doi:10.1007/11917496_29, MR 2290741.
26. ^ On total functions, existence theorems and computational complexity
27. ^ http://www.openproblemgarden.org/?q=op/theoretical_computer_science/subset_sums_equality
28. ^ 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