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 it is also true that if NPI problems exist, then P ≠ NP, 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.[2][3] Some problems that are considered good candidates for being NP-intermediate are the graph isomorphism problem, factoring, and computing the discrete logarithm.[4]

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

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[5]
• The linear divisibility problem[6]

Boolean logic

• Intersecting Monotone SAT[7]
• Minimum Circuit Size Problem[8][9]
• Monotone self-duality[10]

Computational geometry and computational topology

• Computing the rotation distance[11] between two binary trees or the flip distance between two triangulations of the same convex polygon
• The turnpike problem[12] of reconstructing points on line from their distance multiset
• The cutting stock problem with a constant number of object lengths[13]
• Knot triviality[14]
• Deciding whether a given triangulated 3-manifold is a 3-sphere
• Gap version of the closest vector in lattice problem[15]
• Finding a simple closed quasigeodesic on a convex polyhedron[16]

Game theory

• Determining winner in parity games[17]
• Determining who has the highest chance of winning a stochastic game[17]
• Agenda control for balanced single-elimination tournaments[18]

Graph algorithms

• Graph isomorphism problem
• Planar minimum bisection[19]
• Deciding whether a graph admits a graceful labeling[20]
• Recognizing leaf powers and k-leaf powers[21]
• Recognizing graphs of bounded clique-width[22]
• Finding a simultaneous embedding with fixed edges[23]

Miscellaneous

• Assuming NEXP is not equal to EXP, padded versions of NEXP-complete problems
• Problems in TFNP[24]
• Pigeonhole subset sum[25]
• Finding the VC dimension[26]

References

1. Ladner, Richard (1975). "On the Structure of Polynomial Time Reducibility". Journal of the ACM. 22 (1): 155–171. doi:10.1145/321864.321877. S2CID 14352974.
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. Schaefer, Thomas J. (1978). "The complexity of satisfiability problems" (PDF). Proc. 10th Ann. ACM Symp. on Theory of Computing. pp. 216–226. MR 0521057.
4. "Problems Between P and NPC". Theoretical Computer Science Stack Exchange. 20 August 2011. Retrieved 1 November 2013.
5. http://blog.computationalcomplexity.org/2010/07/what-is-complexity-of-these-problems.html
6. https://cstheory.stackexchange.com/q/4331
7. https://cstheory.stackexchange.com/q/1739
8. https://cstheory.stackexchange.com/q/1745
9. 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, S2CID 785205, ECCC TR99-045
10. https://cstheory.stackexchange.com/q/3950
11. Rotation distance, triangulations, and hyperbolic geometry
12. Reconstructing sets from interpoint distances
13. https://cstheory.stackexchange.com/q/3827
14. https://cstheory.stackexchange.com/q/1106
15. https://cstheory.stackexchange.com/q/7806
16. 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.
17. http://kintali.wordpress.com/2010/06/06/np-intersect-conp/
18. https://cstheory.stackexchange.com/q/460
19. Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
20. https://cstheory.stackexchange.com/q/6384
21. 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.
22. 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.
23. 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 (PDF), Lecture Notes in Computer Science, 4271, Berlin: Springer, pp. 325–335, doi:10.1007/11917496_29, MR 2290741.
24. On total functions, existence theorems and computational complexity
25. http://www.openproblemgarden.org/?q=op/theoretical_computer_science/subset_sums_equality
26. 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

External links

• 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.