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 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, 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.[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]
- Factoring integers
- Isomorphism problems: Graph isomorphism problem, Group isomorphism problem, Group automorphism, Ring isomorphism, Ring automorphism
- Computing the rotation distance[5] between two binary trees or the flip distance between two triangulations of the same planar point set
- The Turnpike Problem[6] of reconstructing points on line from distances
- Discrete Log Problem and others related to cryptographic assumptions
- Determining winner in parity games[7]
- Determining who has the highest chance of winning a stochastic game[7]
- Numbers in boxes problems[8]
- Agenda control for balanced single-elimination tournaments[9]
- Knot triviality[10]
- Assuming NEXP is not equal to EXP, padded versions of NEXP-complete problems
- Problems in TFNP[11]
- Intersecting Monotone SAT[12]
- Minimum Circuit Size Problem[13][14]
- Deciding whether a given triangulated 3-manifold is a 3-sphere
- The Cutting Stock Problem with a constant number of object lengths[15]
- Monotone Self-Duality[16]
- Planar Minimum Bisection[17]
- Pigeonhole Subset Sum[18]
- Square Root Sums[19]
- Deciding Whether a Graph Admits a Graceful Labeling[20]
- Gap version of the closest vector in lattice problem[21]
- The linear divisibility problem[22]
- Matching preclusion
- Finding the VC dimension
References
- ↑ Ladner, Richard (1975). "On the Structure of Polynomial Time Reducibility". Journal of the ACM (JACM) 22 (1): 155–171. doi:10.1145/321864.321877.
- ↑ 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.
- ↑ "Problems Between P and NPC". Theoretical Computer Science Stack Exchange. 20 August 2011. Retrieved 1 November 2013.
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/237#237
- ↑ Rotation distance, triangulations, and hyperbolic geometry
- ↑ Reconstructing sets from interpoint distances
- ↑ 7.0 7.1 http://kintali.wordpress.com/2010/06/06/np-intersect-conp/
- ↑ http://blog.computationalcomplexity.org/2010/07/what-is-complexity-of-these-problems.html
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/460#460
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1106#1106
- ↑ On total functions, existence theorems and computational complexity
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1739#1739
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/1745#1745
- ↑ 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
- ↑ http://cstheory.stackexchange.com/questions/3826/np-hardness-of-a-special-case-of-orthogonal-packing-problem/3827#3827
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/3950#3950
- ↑ Approximability of the Minimum Bisection Problem: An Algorithmic Challenge
- ↑ http://www.openproblemgarden.org/?q=op/theoretical_computer_science/subset_sums_equality
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4010#4010
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/6384#6384
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/7806#7806
- ↑ http://cstheory.stackexchange.com/questions/79/problems-between-p-and-npc/4331#4331
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.